# On the cyclic 3-manifold covers of the type Surface $\times$ $\mathbb{R}$

Jordan Sahattchieve

###### Abstract.

This article contains a proof of the fact that, under certain mild technical
conditions, the action of the automorphism group of a cyclic 3-manifold cover
of the type $S\times\mathbb{R}$, where $S$ is a compact surface, yields a
compact quotient. This result is then immediately applied to extend a theorem
in [9] on the fiberings over $\mathbb{S}^{1}$ of certain compact 3-manifolds
which are torus sums. As a corollary, I prove the validity of the conditional
main theorem in [10] and [11]. This paper also furnishes a proof of the
irreducibility of the summands of compact 3-manifolds which are torus sums and
irreducible.

MSC Subject Classifications: 57K30, 57M07, 57M10, 57M60.

In memoriam G. Peter Scott, November 27, 1944 - September 19, 2023

##  1\. Introduction

In this paper I examine the covering type actions of the infinite cyclic group
$\mathbb{Z}$ on the trivial line bundle over a compact surface. I show that
such actions necessarily yield compact quotients under very mild hypotheses.
As a consequence, I prove an extension of Theorem 2.9 in [9] to compact
manifolds with nonempty boundary, and I immediately apply this result to
conclude that the conditional main theorem in [10] and [11] holds. This type
of result is not altogether surprising in light of the Flat Torus Theorem in
CAT(0) geometry. In the context of the Flat Torus Theorem, however, the space
$X$ is assumed to be nonpositively curved and to have a cocompact action of a
group by isometries; the conclusion there is that, if $G$ contains a free
abelian group $A$ of rank $n$, then $X$ is isometrically a product
$Y\times\mathbb{R}^{n}$, and $A$ acts by isometries on $\mathbb{R}^{n}$ with
quotient the flat $n$-torus $\mathbb{T}^{n}$. Here, the space $M$ is a
$3$-manifold which is already known to be a product of a compact $2$-manifold
$S$ and $\mathbb{R}$, and we have an action by a group $G$ which contains an
infinite cyclic group $\mathbb{Z}$. The action is not a priori by isometries,
but by covering transformations. Our conclusion here is that if only finitely
many surfaces in the $\mathbb{Z}$-orbit of any given horizontal section
intersect it, then such an action necessarily produces a compact quotient.
This provides a tool for closing the gap in Theorem 2.10 in [9] for compact
manifolds with nonempty boundary. These investigations yield, as a significant
corollary, a proof of the conditional Theorem 5.3 in [10], as corrected in
[11].

Additionally, in Section 3 below, I prove that whenever an irreducible
$3$-manifold is a torus sum, then each summand is itself an irreducible
$3$-manifold. This result, albeit not surprising, does not, to my knowledge,
appear to exist in written form until now. I am thankful to my late Ph.D.
adviser, Peter Scott, for communicating to me via e-mail a rough idea of using
the torus’s incompressibility and connectedness to write a proof, which was
also our last mathematical conversation.

##  2\. The Main Result

######  Theorem 2.1.

Let $M$ be the $3$-manifold $S\times\mathbb{R}$, which is the trivial line
bundle over the compact surface $S$ with or without boundary, and let $M$ be
orientable. Suppose that there is a covering action of the group $G$ on $M$
and a Riemannian metric on $M$ for which this action is by isometries, and
suppose that $G$ contains a subgroup $\left\langle\gamma\right\rangle<G$, with
$\gamma\in G$, isomorphic to the infinite cyclic group $\mathbb{Z}$. If
$\gamma^{n}S_{0}\cap S_{0}=\emptyset$, for all $n>n_{0}$, where $S_{0}$ is any
horizontal section $S_{0}=S\times\left\\{t_{0}\right\\}$ with
$t_{0}\in\mathbb{R}$, and $n_{0}$ is a positive integer, then the quotient of
$M$ by the action of $G$ is compact.

###### Proof.

Let $g=\gamma^{n_{0}+1}\in G$. First, note that $g^{i}S_{0}\cap
S_{0}=\emptyset$, for all $i>0$. Note, also, that if $g^{-i}S_{0}\cap
S_{0}\neq\emptyset$, for some $i>0$, then $g^{i}(g^{-i}S_{0}\cap
S_{0})\neq\emptyset$, and therefore $S_{0}\cap g^{i}S_{0}\neq\emptyset$, hence
$i=0$. Thus, we conclude that $g^{i}S_{0}\cap S_{0}=\emptyset$, for all $i\neq
0$.

We will show that the translates $g^{i}S_{0}$ partition $M$ into compact
connected submanifolds which meet only at their boundaries. To do this, note
that since $S_{0}$ separates $M$ into two connected components, so does
$gS_{0}$. Let the closures of the two connected components of $M-S_{0}$ be
$V^{+}$ and $V^{-}$. Then, the closures of the connected components of
$M-gS_{0}$ are $gV^{+}$ and $gV^{-}$, respectively. Since $S_{0}\cap
gS_{0}=\emptyset$, we must have $gS_{0}\subset(V^{+})^{0}$ or
$gS_{0}\subset(V^{-})^{0}$. Suppose, without loss of generality, that
$gS_{0}\subset(V^{+})^{0}$. Since $S_{0}$ disconnects $M$ into exactly two
connected components, $gS_{0}$ must disconnect $V^{+}$ into exactly two
connected components also, since $M-gS_{0}=(V^{+}-gS_{0})\cup V^{-}$ and
$S_{0}\cap gS_{0}=\emptyset$. One of the two connected components of
$M-gS_{0}$ must contain $S_{0}$. Let the closure of this connected component
be denoted by $D$; it is now clear that $\partial D=S_{0}\cup gS_{0}$. We will
show that $D$ is a connected submanifold of $M$ whose boundary is $S_{0}\cup
gS_{0}$, that $D$ is compact, that $g^{i}D\cap g^{j}D=\emptyset$ for $i\neq
j$, unless $|i-j|=1$ in which case $g^{i}D\cap g^{j}D$ is the translate of
$S_{0}$ which is the common boundary surface of both $g^{i}D$ and $g^{j}D$,
and that $M=\bigcup g^{k}D$:

$D$ is compact: Let $h:M\rightarrow\mathbb{R}$ be the projection to the
$\mathbb{R}$ factor. If necessary, we can compose $h$ with a linear map, so
that $h(x)>t_{0}$ for $x\in gS_{0}$ and $h(V^{+})\subset\mathbb{R}_{\geq 0}$.
Since $gS_{0}$ is compact and connected, $h(gS_{0})\subset\mathbb{R}$ is a
closed interval; let $h_{max}$ be its upper bound. Let
$x_{1}=(s_{1},t_{1}),x_{2}=(s_{2},t_{2})\in D$ be any two points. Consider the
vertical lines $\left\\{s_{1}\right\\}\times\mathbb{R}$ and
$\left\\{s_{2}\right\\}\times\mathbb{R}$. Because $gS_{0}$ disconnects
$V^{+}$, $\left\\{s_{i}\right\\}\times\mathbb{R}$ must intersect $gS_{0}$ in
at least one point which is at a distance at most $h_{max}-t_{0}$ away from
$x_{i}$. Therefore, $x_{1}$ can be joined to $x_{2}$ by a path of length no
more than $2(h_{max}-t_{0})+diam(gS_{0})$. Since $gS_{0}$ is compact, this
number is not $\infty$, and we conclude that $D$ is a closed bounded set. By
the theorem of Hopf-Rinow, see Theorem 2.8(b) in [1], $D$ is compact.

$g^{2}S_{0}\subset V^{+}-D$: We must have either $g^{2}S_{0}\subset V^{+}-D$,
or $g^{2}S_{0}\subset D$, or $g^{2}S_{0}\subset V^{-}$. Let $W^{+}$ be the
closure of the connected component of $M-gS_{0}$ which contains $D$, and let
$W^{-}$ be the closure of its complement. If $g^{2}S_{0}\subset D$, then
$gW^{+}\subset W^{+}$ or $gW^{+}\subset V^{+}$; however, since
$(W^{+})^{0}\supset S_{0}$, hence $(gW^{+})^{0}\supset gS_{0}$, thus if
$gW^{+}\subset W^{+}$, we would have $gS_{0}\subset(W^{+})^{0}$, which is not
true. Therefore, we have $gW^{+}\subset V^{+}$. On the other hand,
$gV^{+}=W^{+}$ or $gV^{+}=W^{-}$. Since $D=V^{+}\cap W^{+}$, we have
$gD=gV^{+}\cap gW^{+}$, hence must have $gD=W^{+}\cap gW^{+}$, or
$gD=W^{-}\cap gW^{+}$. Now, $W^{-}\cap g^{2}S_{0}=\emptyset$, thus $W^{-}$ is
contained in one of $(gW^{+})^{0}$ or $(gW^{-})^{0}$. Since
$(W^{+})^{0}\supset S_{0}$, we have $(gW^{+})^{0}\supset gS_{0}$, and
$(gW^{+})^{0}\cap W^{-}\neq\emptyset$, hence $W^{-}\subset gW^{+}$. This
imples that if $gD=W^{-}\cap gW^{+}$, then $gD$ would fail to be compact since
$gD\supset W^{-}$, which is not true. Therefore, $gD=W^{+}\cap gW^{+}$, in
which case we also have $gD\subset D$, as $gW^{+}\subset V^{+}$. We must also
have $gD\neq D$, since $D$ and $gD$ have different boundaries, i.e. $gD$ is
properly contained in $D$, with $D-gD$ having nonempty interior. Since $M$ is
orientable, the Riemannian metric for which $G$ acts by isometries induces a
nowhere vanishing volume form $\omega$ on $M$, such that $G$ preserves the
Riemannian measure induced by $\omega$. But since $gD\subset D$, we must have
$vol(gD)<vol(D)$, which is impossible as we must have $vol(gD)=vol(D)$.

Similarly, if $g^{2}S_{0}\subset V^{-}$, let $D^{\prime}$ be the closure of
the connected component of $V^{-}-g^{2}S_{0}$ which contains $S_{0}$: it is a
connected submanifold of $M$ whose boundary is $S_{0}\cup g^{2}S_{0}$. Let
$W^{+}$ be the component of $M-gS_{0}$ which contains $D\cup D^{\prime}$, and
similarly let $Q^{+}$ be the component of $M-g^{2}S_{0}$ which contains $D\cup
D^{\prime}$. With this notation, we have $W^{+}=D\cup V^{-}$, and
$Q^{+}=D^{\prime}\cup V^{+}$. It is now straightforward to verify that $D\cup
D^{\prime}=W^{+}\cap Q^{+}$, and therefore $g(D\cup D^{\prime})=gW^{+}\cap
gQ^{+}$. Since the isometry $x\rightarrow g^{-1}x$ takes $g^{2}S_{0}$ to
$gS_{0}$ and $gS_{0}$ to $S_{0}$, we must have (a.) $gW^{+}=V^{+}$ or
$gW^{+}=V^{-}$, and (b.) $gQ^{+}=W^{-}\subset(V^{+})^{0}$ or $gQ^{+}=W^{+}$.
In the cases (1.) $gW^{+}=V^{+}$ and $gQ^{+}\subset V^{+}$, and (2.)
$gW^{+}=V^{-}$ and $gQ^{+}=W^{+}$, the intersection $gW^{+}\cap gQ^{+}$ is
noncompact, and therefore cannot equal $g(D\cup D^{\prime})$. In the case (3.)
$gW^{+}=V^{-}$ and $gQ^{+}\subset(V^{+})^{0}$, the intersection $gW^{+}\cap
gQ^{+}$ is empty, while case (4.) $gW^{+}=V^{+}$ and $gQ^{+}=W^{+}$, $g(D\cup
D^{\prime})=D$ is seen to be impossible by a volume comparison argument as
above. Therefore, $g^{2}S_{0}\subset V^{+}-D$. Now, we see that $D$ and $gD$
share the boundary surface $gS_{0}$, and if $D^{0}\cap(gD)^{0}\neq\emptyset$,
then $D-gS_{0}$ will remain connected while having nontrivial intersection
with both of the connected components of $M-gS_{0}$, which is impossible.

Let $D_{k}=\bigcup_{i=0}^{k}g^{i}D_{k}$. Then, $D_{k}$ is connected,
$D_{k}\subset V^{+}$, $g^{k+1}S_{0}\subset V^{+}-D_{k}$, and $g^{k+1}D\cap
D_{k}=g^{k}S_{0}$. We proceed by induction on $k$ to show that $g^{k+1}S_{0}$
cannot be contained in either $V^{-}$ or $D_{k-1}$.

Let $W^{+}$ be the closure of the connected component of $M-g^{k}S_{0}$ which
contains $D_{k-1}$. Since $g^{k+1}S_{0}$ is disjoint from all $g^{i}S_{0}$,
for $0\leq i\leq k$, if $g^{k+1}S_{0}\subset D_{k-1}$, then
$g^{k+1}S_{0}\subset g^{j}D^{0}$, for some $0\leq j\leq k-1$. By multiplying
by $g^{-j}$, this is equivalent to $g^{k-j+1}S_{0}\subset D^{0}$, and since
$D^{0}\subset D_{k-j}$, we have $g^{k-j+1}S_{0}\subset D_{k-j}$, which is
ruled out by the induction hypothesis if $j>0$. Therefore, we now assume that
$j=0$, hence that $g^{k+1}S_{0}\subset D^{0}$. In this case, $g^{k+1}S_{0}$
separates $D$ into two connected components, the closures of which we denote
by $E$ and $F$. To be specific, $E$ is the closure of the connected component
of $D-g^{k+1}S_{0}$ bounded by $S_{0}$ and $g^{k+1}S_{0}$, while $F$ is the
closure of the component bounded by $gS_{0}$ and $g^{k+1}S_{0}$. Since
$gW^{+}$ is the closure of a connected component of $M-g^{k+1}S_{0}$, we
immediately see that either $gW^{+}=E\cup V^{-}$ or $gW^{+}=cl(V^{+}-E)$. On
the other hand, $gV^{+}$ is the closure of a connected component of
$M-gS_{0}$, thus we must have $gV^{+}=D\cup V^{-}$ or $gV^{+}=cl(V^{+}-D)$.
Since $D_{k-1}=V^{+}\cap W^{+}$, we have $gD_{k-1}=gV^{+}\cap gW^{+}$. In the
cases $gW^{+}=E\cup V^{-}$, $gV^{+}=D\cup V^{-}$, and $gW^{+}=cl(V^{+}-E)$,
$gV^{+}=cl(V^{+}-D)$, the intersection $gV^{+}\cap gW^{+}$ fails to be
compact, hence cannot equal $gD_{k-1}$. In the case $gW^{+}=E\cup V^{-}$ and
$gV^{+}=cl(V^{+}-D)$, we have $gV^{+}\cap gW^{+}=\emptyset$, which is again a
contradiction. In the remaining case, $gW^{+}=cl(V^{+}-E)$ and $gV^{+}=D\cup
V^{-}$, we have $gD_{k-1}=F\subset D$, which is ruled out by a volume
argument.

Now, if $g^{k+1}S_{0}\subset V^{-}$, let $D^{\prime}$ be the closure of the
component of $V^{-}-g^{k+1}S_{0}$ which contains $S_{0}$ \- it is a connected
submanifold bounded by $S_{0}$ and $g^{k+1}S_{0}$. Let $W^{+}$ be as above,
and let $Q^{+}$ now be the connected component of $M-g^{k+1}S_{0}$ which
contains $D_{k-1}\cup D^{\prime}$. We now have $D_{k-1}\cup
D^{\prime}=W^{+}\cap Q^{+}$, and therefore
$g^{-1}(D_{k-1}\cup D^{\prime})=g^{-1}W^{+}\cap g^{-1}Q^{+}$. Using the same
arguments as before, we see that either $g^{-1}W^{+}=D_{k-2}\cup V^{-}$, or
$g^{-1}W^{+}=W^{-}\cup g^{k-1}D$, and that we also have either
$g^{-1}Q^{+}=W^{+}$, or $g^{-1}Q^{+}=W^{-}$. In either of the two cases
$g^{-1}W^{+}=D_{k-2}\cup V^{-}$, $g^{-1}Q^{+}=W^{+}$, and
$g^{-1}W^{+}=W^{-}\cup g^{k-1}D$, $g^{-1}Q^{+}=W^{-}$, the intersection
$g^{-1}W^{+}\cap g^{-1}Q^{+}$ fails to be compact, as it contains the
noncompact set $V^{-}$, or $W^{-}$, respectively. In the case
$g^{-1}W^{+}=W^{-}\cup g^{k-1}D$, $g^{-1}Q^{+}=W^{-}$, we have
$g^{-1}W^{+}\cap g^{-1}Q^{+}=\emptyset$, which is a contradiction. The last
case to consider is $g^{-1}W^{+}=W^{-}\cup g^{k-1}D$ and $g^{-1}Q^{+}=W^{+}$.
Here, $g^{-1}W^{+}\cap g^{-1}Q^{+}=g^{k-1}D$, which gives $g^{-1}(D_{k-1}\cup
D^{\prime})=g^{k-1}D$; a contradiction follows from a volume comparison
argument, after noting that $k>0$.

Thus, we conclude that $g^{k+1}S_{0}\subset V^{+}-D_{k-1}$, and also that the
connected submanifold $g^{k}D$ is contained in $V^{+}$, hence
$D_{k}=D_{k-1}\cup g^{k}D\subset V^{+}$.

Given any $x_{0}\in V^{+}$, there exists $k$ such that $x\in D_{k}$.
Let $l=inf\left\\{d(x,y):x\in S_{0},y\in gS_{0}\right\\}$ be the distance
between $S_{0}$ and $gS_{0}$ in the path metric induced by the Riemannian
metric on $M$. By construction of the submanifold $D_{k}$, we see that any
path from $g^{k}S_{0}$ to $S_{0}$ must pass through all $g^{i}S_{0}$ for
$1\leq i\leq k-1$, hence $l_{k}=\inf\left\\{d(x,y):x\in g^{k}S_{0},y\in
S_{0}\right\\}=kl$ is an increasing function of $k$. Suppose now that
$x_{0}\notin D_{k}$ for all $k$, and choose $k_{0}$ large enough so that
$l_{k_{0}}=k_{0}l>d(x_{0},S_{0})$. Since $k_{0}\notin D_{k_{0}}$, any path
from $x_{0}$ must pass through all the $g^{i}S_{0}$ for $1\leq i\leq k$ as all
of them disconnect $M$. Then, we must have $d(x_{0},S_{0})\geq
d(g^{k_{0}}S_{0},S_{0})=k_{0}l$ which is a contradiction.

$g^{-1}S_{0}\subset V^{-}$: We argue by contradiction. If $g^{-1}S_{0}\subset
V^{+}$, then since $V^{+}=\bigcup_{i\geq 0}g^{i}D$, and since the $g^{j}S_{0}$
are disjoint for any two different values of $j$, we must have
$g^{-1}S_{0}\subset(g^{k}D)^{0}$ for some $k\geq 0$. Then,
$S_{0}\subset(g^{k+1}D)^{0}$, where $k+1\geq 1$, thus $g^{k+1}S_{0}$ and
$g^{k+2}S_{0}$ would lie in two different connected components of $M-S_{0}$,
whereas we know that this is not the case, as $g^{i}S_{0}\subset(V^{+})^{0}$
for all $i>0$.

By an analogous argument, $V^{-}=\bigcup_{i<0}g^{i}D$. This shows the
cocompactness of the action of $G$ on $M$. ∎

I shall now apply Theorem 2.1 to prove a result which strengthens Theorem 2.9
in [9], and which by doing so closes the gap in the proof of Theorem 2.10 in
[9] for manifolds with nonempty boundary. I shall also prove that a suitably
chosen finite cover of $M$ fibers in the desired way, meaning that
$\pi_{1}(F)$ is commensurable with $U$. This proof covers the case $\partial
M\neq\emptyset$ as well as the case $\partial M=\emptyset$, neither of which
has been established in [9]. While this does not prove commensurability for
the particular cover of $M$ considered in [9], it does establish the fibration
result claimed therein and also in this article.

Before I state the theorem, for ease of reference, I shall restate a certain
technical condition regarding the 3-manifolds under consideration here and in
[2], [9], and [10]:

(A) Let $U$ be a finitely generated subgroup of the fundamental group $G$ of a
compact 3-manifold $M$. Suppose that $U$ contains a nontrivial subnormal
subgroup $N$ of $G$, and that the index $|G:U|$ is infinite. If $N$ is not
isomorphic to the infinite cyclic group $\mathbb{Z}$, then $M$ is finitely
covered by a bundle over $\mathbb{S}^{1}$ with fiber a compact surface $F$ and
$\pi_{1}(F)$ is commensurable with $U$.

The promised result is the following theorem:

######  Theorem 2.2.

Let $M$ be a compact 3-manifold with $M=X_{1}\cup_{T}X_{2}$, or
$M=X_{1}\cup_{T}$, where statement (A) holds for $X_{i}$, with $i=1,2$.
Suppose that $G=\pi_{1}(M)$ contains a finitely generated subgroup $U$ of
infinite index satisfying the following:

  1. (1)

$U$ contains a nontrivial subnormal subgroup $N$ of $G$,

  2. (2)

$N$ intersects nontrivially the fundamental group of the splitting torus,

  3. (3)

$N\cap\pi_{1}(X_{i})$ is not isomorphic to $\mathbb{Z}$.

If the graph of groups $\mathcal{U}$ corresponding to $U$ is of finite
diameter, then $\widehat{M}$ has a finite cover $\widetilde{M}$ which is a
bundle over $\mathbb{S}^{1}$ with fiber a compact surface $F$, and
$\pi_{1}(F)$ is commensurable with $U$.

###### Proof.

If $M$ is closed, the theorem is essentially a restatement of Theorem 2.9 in
[9], with the trivial consideration that $N$ is subnormal rather than normal.
Therefore I now assume that $\partial M\neq\emptyset$. The reader can verify
for themselves that the proof of Theorem 2.9 in [9] remains valid up to line 2
on page 31 under the hypothesis on $N$ and $M$ stated above when $\partial
M\neq\emptyset$. The only difference is that the surface $S^{\prime}$
constructed in the proof of Theorem 2.9 is now a surface with nonempty
boundary, and consequently the manifold $M^{\prime\prime}$, the compact core
of $M^{\prime}$, is itself a manifold with nonempty boundary. While the
remainder of the proof of Theorem 2.9 fails when $\partial M\neq\emptyset$, we
can apply Theorem 2.1 after showing that we have an action of $\mathbb{Z}$ on
the cover $M=S^{\prime}\times\mathbb{R}$, which satisfies the hypothesis of
the theorem above. Using the notation employed in [9], we see that we have the
tower of covers $S^{\prime}\times\mathbb{R}\rightarrow M^{\prime}\rightarrow
M$.

We verify that the ingredients in the proof in [9] are also present in the
$\partial M\neq\emptyset$ case, although for different reasons. We first show
that $\pi_{1}(S^{\prime})\neq\mathbb{Z}$, from which it will follow that
$N_{\pi_{1}(S^{\prime})}=\pi_{1}(M^{\prime})$ is finitely generated.

Case 1: $M=X_{1}\cup_{T}X_{2}$

First, since $S^{\prime}\times\mathbb{R}$ is an orientable cover of the
orientable manifold $M$, it is itself orientable, which, in turn, implies that
the submanifold $S^{\prime}\times\left[-1,1\right]$ of
$S^{\prime}\times\mathbb{R}$ is orientable, hence $S^{\prime}$ is itself
orientable, by the remark on lines 7 and 6 from the bottom, on page 5 of [5].
We also observe that $S^{\prime}$ was obtained by gluing finitely many copies
of the surfaces $S_{i}$, which are finite covers of the fibers denoted in [9]
by $F_{i}$ chosen in such a way so that $\partial S_{i}$ has at least 2
boundary components. If either $S_{i}$ were a surface other than the annulus
$\mathbb{S}^{1}\times I$, then $\pi_{1}(S_{i})$ would be a free group of rank
at least 2. Since $\pi_{1}(S^{\prime})$ has the structure of a graph of groups
where the vertex groups are all isomorphic to $\pi_{1}(S_{i})$, which must
necessarily embed in $\pi_{1}(S^{\prime})$ by Bass-Serre theory, we conclude
$\pi_{1}(S^{\prime})$ would contain a free group of rank at least 2, which is
clearly impossible. Therefore, both $S_{1}$ and $S_{2}$ are the annulus
$\mathbb{S}^{1}\times I$. However, in this case, $X_{i}$ is virtually covered
by an annulus bundle over $\mathbb{S}^{1}$, and thus $\pi_{1}(X_{i})$ contains
$\mathbb{Z}\rtimes\mathbb{Z}$ with finite index. However, such a group cannot
contain a (finitely generated) subgroup of infinite index $U$, which contains
a nontrivial normal subgroup $N$ not isomorphic to the infinite cyclic group
$\mathbb{Z}$, which is a contradiction. Hence, again by Theorem 3.2 [14],
$N_{\pi_{1}(S^{\prime})}$ is finitely generated. As in the case without
boundary, $\pi_{1}(T)\cap\pi_{1}(S^{\prime})$ is isomorphic to $\mathbb{Z}$:
It is nontrivial as in [9], since the boundary components of the copies of
$S_{i}$ represent elements of $\pi_{1}(S_{i})$ which map to a nontrivial
$\alpha^{r_{1}r_{2}}\in\pi_{1}(M)$; thus, $\pi_{1}(T)\cap\pi_{1}(S^{\prime})$
a nontrivial free group, which is a subgroup of a free abelian group. We again
conclude that as in [9],
$|N_{\pi_{1}(S^{\prime})}:\pi_{1}(S^{\prime})|=\infty$.

Case 2: $M=X_{1}\cup_{T}$

The argument in this case is analogous, as $S^{\prime}$ remains a surface with
nonempty boundary after the gluing of the boundary annuli of
$F_{1}\times\mathbb{R}$ which map to the boundary tori $T_{i}$ in $Y_{1}$.
Therefore, again $\pi_{1}(S^{\prime})=\mathbb{Z}$ implies that $S^{\prime}$ is
the annulus: In this case, $S^{\prime}$ was obtained by gluing boundary
components of $F_{1}$; therefore,
$\pi_{1}(S^{\prime})=\left(\left(\left(\pi_{1}(F_{1})*_{\mathbb{Z}}\right)*_{\mathbb{Z}}\right)...\right)*_{\mathbb{Z}}$,
and $F_{1}$ must be the annulus by the same rank argument, hence because
$\partial S^{\prime}\neq\emptyset$, we have $F_{1}=S^{\prime}$. As in the case
above, this implies a contradiction with the theorem’s hypothesis. Once again,
$|N_{\pi_{1}(S^{\prime})}:\pi_{1}(S^{\prime})|=\infty$.

Now, applying Theorem 3 in [7], we conclude that a finite cover of
$\widehat{M^{\prime\prime}}$, the Poincaré associate of the core of
$M^{\prime}$, fibers over $\mathbb{S}^{1}$ with fiber a compact surface, and
that $\pi_{1}(S^{\prime})$ is subgroup of finite index of the fundamental
group of the fiber; thus, $\pi_{1}(M^{\prime})=\pi_{1}(M^{\prime\prime})$
contains $\pi_{1}(S^{\prime})$ as a normal subgroup, the quotient by which
contains $\mathbb{Z}$ with finite index.

Hence, we see that the automorphism group $\mathfrak{G}$ of the cover
$S^{\prime}\times\mathbb{R}\rightarrow M^{\prime}$, being isomorphic to the
quotient of the normalizer of $\pi_{1}(S^{\prime}\times\mathbb{R})$ in
$\pi_{1}(M^{\prime})$ by $\pi_{1}(S^{\prime}\times\mathbb{R})$ \- see
Proposition 1.39(b) in [6], contains the infinite cyclic group $\mathbb{Z}$
with finite index.

The manifold $M$ is compact, therefore, we can take its double $DM$, embed it
in $\mathbb{R}^{W}$ for some $W\in\mathbb{N}$ by Whitney’s Embedding Theorem,
see page 53 in [8], and pull back the Riemannian metric on $\mathbb{R}^{W}$ to
obtain a Riemannian metric on $M$, which we can then pull back on
$M^{\prime}$, and then on $S^{\prime}\times\mathbb{R}$, under the covering
maps to obtain the desired Riemannian metric on $S^{\prime}\times\mathbb{R}$
for which the action of $\mathfrak{G}$ is by isometries.

The final step we need to make, before we can apply Theorem 2.1, is to show
that for $S_{0}=S^{\prime}\times\left\\{0\right\\}$, we can find a positive
integer $n_{0}$ with the property that $\gamma^{n}S_{0}\cap S_{0}=\emptyset$,
for all $n\geq n_{0}$, where $\gamma$ is a generator of the infinite cyclic
subgroup of the automorphism group $\mathfrak{G}$.

First, we produce an integer $n_{0}>0$ such that $\gamma^{n}S_{0}\cap
S_{0}\neq S_{0}$, for all $n\geq n_{0}$. Note that if for some $n$ we have
$\gamma^{n}S_{0}\cap S_{0}=S_{0}$, then $\gamma^{n}S_{0}\supset S_{0}$, and if
additionally $\gamma^{n}S_{0}\neq S_{0}$, then
$S^{\prime}\times\mathbb{R}-S_{0}$ will remain connected, as
$S^{\prime}\times\mathbb{R}-S_{0}\supset(S^{\prime}\times\mathbb{R}-\gamma^{n}S_{0})\cup\left\\{p\right\\}$,
for some $p\in\gamma^{n}S_{0}-S_{0}$, and we see that the set on the right
side of the containment is connected. Therefore, for this $n$ we have
$\gamma^{n}S_{0}=S_{0}$. This cannot hold for infinitely many values of $n>0$,
because $S_{0}$ is compact. To show this, we argue by contradiction and we
make the assumption that the set of $n$ such that $\gamma^{n}S_{0}=S_{0}$ is
infinite. Let $n_{i}$ be an infinite sequence of such values of $n$. Then,
since $\left\langle\gamma\right\rangle$ acts by covering transformations,
given any $x_{0}\in S_{0}$, there is an open set $\mathcal{U}\subset
S^{\prime}\times\mathbb{R}$, such that if
$\gamma^{i}\mathcal{U}\cap\gamma^{j}\mathcal{U}\neq\emptyset$, then $i=j$, see
page 72 in [6]. But we can find, for the length metric induced by the
Riemannian metric on $S^{\prime}\times\mathbb{R}$ for which $\eta$ is an
isometry, an open ball $B(x_{0},\epsilon)\subset\mathcal{U}$, and we see that
$d(\gamma^{i}x_{0},\gamma^{j}x_{0})>2\epsilon$, for any $i\neq j$. Therefore,
the sequence $\gamma^{n_{i}}x_{0}\in S_{0}$ can have no Cauchy subsequence and
no accumulation point in $S_{0}$, which it must, by compactness of $S_{0}$.
Hence, there exists an integer $n_{0}>0$ such that for all $n>n_{0}$,
$\gamma^{n}S_{0}\neq S_{0}$. From now on, we let $n$ be an integer greater
than $n_{0}$, so that $\gamma^{n}S_{0}\cap S_{0}\neq S_{0}$. We proceed to
show that we must also have $\gamma^{n}S_{0}\cap S_{0}=\emptyset$.

Now, recall from [9] that the cover $S^{\prime}\times\mathbb{R}$ is obtained
by gluing copies of $S_{i}\times\mathbb{R}$ along their boundary annuli which
are of the form $C_{i}\times\mathbb{R}$, where $C_{i}$ is a component of
$\partial S_{i}$ which projects under $p_{1}\circ p_{2}$ and $q_{1}\circ
q_{2}$, respectively, into $T$. The covering map
$\eta:S^{\prime}\times\mathbb{R}\rightarrow M$ is described by pasting
together the various copies of the covering maps $p_{1}\circ p_{2}\circ p_{3}$
and $q_{1}\circ q_{2}\circ q_{3}$ to $X_{1}$ and $X_{2}$ from the copies of
$S_{i}\times\mathbb{R}$. If $\gamma^{n}S_{0}\cap S_{0}\neq\emptyset$, then
since $S_{0}$ and $\gamma^{n}S_{0}$ are closed sets, we can find a non-
manifold point $x_{0}\in\gamma^{n}S_{0}\cap S_{0}$ for the union of the two
surfaces $\gamma^{n}S_{0}\cup S_{0}\subset S^{\prime}\times\mathbb{R}$, which
under the local diffeomorphism $\eta$ would map to the non-manifold point
$\eta(x_{0})$ of $\eta((\gamma^{n}S_{0}\cup
S_{0})\cap\eta^{-1}(X_{i})))\subset X_{i}$. Hence, the image of $x_{0}$ under
the appropriate copy of $p_{3}\circ p_{2}$, or $q_{3}\circ q_{2}$, must be a
non-manifold point of a copy of the fiber $F_{i}$ in $Y_{i}$ for $i=1$ or
$i=2$, as both $\gamma^{n}S_{0}\cap\eta^{-1}(X_{i})$ and
$S_{0}\cap\eta^{-1}(X_{i})$ cover the same embedded copy of the fiber $F_{i}$
in $Y_{i}$, and $Y_{i}$ covers $X_{i}$ by a local diffeomorphism. However, all
of the fibers $F_{i}$ of the bundle $Y_{i}$ are embedded surfaces and have no
non-manifold points, which is a contradiction.

Now, by Theorem 2.1, the cover $M^{\prime}$ is compact, since the automorphism
group of the cover $\mathfrak{G}$ acts with compact quotient. Now, we see that
by Theorem 3 in [7], a finite cover $V$ of $\widehat{M^{\prime}}$, and
therefore also of $\widehat{M}$, fibers over the circle with fiber a compact
surface $F$. We now show that either $U$ is commensurable with $\pi_{1}(F)$,
or a finite cover of $V$, hence also of $M$, fibers over the circle with fiber
a compact surface $F^{\prime}$, and $\pi_{1}(F^{\prime})$ is commensurable
with $U$; thus proving, in both cases, that a finite cover of $M$ fibers in
the desired way. To this end, consider $U\cap\pi_{1}(F)<\pi_{1}(M)$. We note
that we have the short exact sequence
$1\rightarrow\pi_{1}(F)\rightarrow\pi_{1}(V)\rightarrow\mathbb{Z}\rightarrow
1$, and that $\pi_{1}(V)$ is a subgroup of finite index of $\pi_{1}(M)$. If
$U\cap\pi_{1}(F)$ is of finite index in both $U$ and $\pi_{1}(F)$, then we are
done - $U$ is commensurable with $\pi_{1}(F)$. Note that if $\pi_{1}(F)\cap
N=\left\\{1\right\\}$, then $\pi_{1}(V)\cap N$ embeds in the infinite cyclic
group $\mathbb{Z}$, which means that $N$ contains $\mathbb{Z}$ with finite
index. In this case, $N$ is a nontrivial finitely generated subnormal subgroup
of $G$, hence by Corollary 2.3 in [2], $N$ is the fundamental group of a
compact surface - see also Theorem 1 in [7]. Let $\Sigma$ be such a surface.
We see that $\partial\Sigma=\emptyset$ is ruled out by the classification of
surfaces, whereas if $\partial\Sigma\neq\emptyset$, then as before, we see,
after passing to the orientable double cover if necessary, that the finite
cover of $\Sigma$ whose fundamental group is $\mathbb{Z}$ is the annulus
$\mathbb{S}^{1}\times I$, hence $\Sigma$ is the annulus or the Moebius strip,
and in either case $N=\mathbb{Z}$, contradicting the hypothesis of the
theorem. Thus, we shall henceforth assume that $\pi_{1}(F)\cap
N\neq\left\\{1\right\\}$.

We consider the remaining cases separately below:

$U\cap\pi_{1}(F)$ is of finite index in $\pi_{1}(F)$ but not in $U$: This is
impossible, since if $\pi_{1}(F)$ were virtually contained in $U$, but $U$
were not virtually contained in $\pi_{1}(F)$, then $U$ would be of finite
index in $\pi_{1}(M)$, which contradicts the theorem’s hypothesis. Here are
the details: Consider the subgroup $(U\cap\pi_{1}(V))\pi_{1}(F)$ of
$\pi_{1}(V)$. The map $(U\cap\pi_{1}(V))f\rightarrow U\cap\pi_{1}(F)f$,
$f\in\pi_{1}(F)$, from the right cosets of $U\cap\pi_{1}(V)$ in
$(U\cap\pi_{1}(V))\pi_{1}(F)$ to the set of right cosets of $U\cap\pi_{1}(F)$
in $\pi_{1}(F)$ is injective, and therefore
$|(U\cap\pi_{1}(V))\pi_{1}(F):U\cap\pi_{1}(V)|<\infty$. Now, consider the
epimorphism $\phi:\pi_{1}(V)\rightarrow\mathbb{Z}$ and its restriction
$\phi|_{U\cap\pi_{1}(V)}:U\cap\pi_{1}(V)\rightarrow\mathbb{Z}$. Since
$|U:U\cap\pi_{1}(V)|<\infty$ and $|U:U\cap\pi_{1}(F)|=\infty$, we must have
$|U\cap\pi_{1}(V):U\cap\pi_{1}(F)|=\infty$. Since now
$Ker(\phi_{U\cap\pi_{1}(V)})=U\cap\pi_{1}(F)$ is of infinite index in
$U\cap\pi_{1}(V)$, we have $\overline{t}^{k}\in Im(\phi|_{U\cap\pi_{1}(V)})$,
for some $k>0$. Thus $t^{k}f\in U\cap\pi_{1}(V)$, for some $f\in\pi_{1}(F)$,
and $t^{k}\in(U\cap\pi_{1}(V))\pi_{1}(F)$. Now,
$(U\cap\pi_{1}(V))\pi_{1}(F)t^{i}\supset\pi_{1}(F)t^{i}$, and since
$\pi_{1}(V)=\bigcup_{i}\pi_{1}(F)t^{i}$, we see that
$\pi_{1}(V)=\bigcup_{i}(U\cap\pi_{1}(V))\pi_{1}(F)t^{i}=\bigcup_{i=0}^{k-1}(U\cap\pi_{1}(V))\pi_{1}(F)t^{i}$.
This shows that $|\pi_{1}(V):(U\cap\pi_{1}(V))\pi_{1}(F)|<\infty$, thus
$|\pi_{1}(V):U\cap\pi_{1}(V)|<\infty$; from here, we get
$|\pi_{1}(M):U\cap\pi_{1}(V)|<\infty$, and finally $|\pi_{1}(M):U|<\infty$ \-
a contradiction, as claimed.

$U\cap\pi_{1}(F)$ is of finite index in $U$, but not in $\pi_{1}(F)$: In this
case, $U\cap\pi_{1}(F)$ is a finitely generated subgroup of $\pi_{1}(F)$ of
infinite index, which contains a nontrivial subnormal subgroup
$N\cap\pi_{1}(F)\triangleleft_{s}\pi_{1}(F)$. By Theorem 2.1 in [10], $F$ is
the torus, since $V$ and therefore $F$ is orientable. However, torus bundles
over $\mathbb{S}^{1}$ are geometric, hence $V$ is a geometric manifold. Since
$U\cap\pi_{1}(V)$ is of finite index in $U$ and of infinite index in
$\pi_{1}(V)$, it is a finitely generated infinite index subgroup of
$\pi_{1}(V)$, whereas the subgroup $N\cap\pi_{1}(V)<U\cap\pi_{1}(V)$ is a
nontrivial subnormal subgroup of $\pi_{1}(V)$ which is not infinite cyclic.
Now, we can apply Theorem 3.9 in [10] to $V$, $U\cap\pi_{1}(V)$, and
$N\cap\pi_{1}(V)$, to conclude that a finite cover of $V$, hence also of $M$,
is a bundle over $\mathbb{S}^{1}$, with fiber a compact surface $F^{\prime}$,
and also that $U\cap\pi_{1}(V)$ is commensurable with $\pi_{1}(F^{\prime})$.
Since $U\cap\pi_{1}(V)$ is of finite index in $U$, we conclude that $U$ is
itself commensurable with $\pi_{1}(F^{\prime})$, as desired.

$U\cap\pi_{1}(F)$ is of finite index in neither $\pi_{1}(F)$ nor $U$: If
$U\cap\pi_{1}(F)$ is finitely generated, then as in the case above, the
conclusion follows from Theorem 3.9 in [10] for the same reasons. Therefore,
we assume that the 3-manifold group $U\cap\pi_{1}(F)$ is not finitely
generated. In such a case, however, Theorem 2.6 in [2] implies that the
nontrivial subnormal subgroup $\pi_{1}(F)\cap N$ of $U\cap\pi_{1}(F)$ is
infinite cyclic and also that $\pi_{1}(F)\cap N\triangleleft U\cap\pi_{1}(F)$.
Since the non-finitely generated group $U\cap\pi_{1}(F)$ properly contains its
infinite cyclic subgroup $\pi_{1}(F)\cap N$, we can find an element $t$ of
$U\cap\pi_{1}(F)$ such that $t\notin\pi_{1}(F)\cap N$. Consider now the
subgroup $H=\left\langle\alpha,t\right\rangle$ of $\pi_{1}(F)$, where $\alpha$
is a generator of $\pi_{1}(F)\cap N=\mathbb{Z}$. Since $H<\pi_{1}(F)$ contains
$\mathbb{Z}^{2}$ with index at most 2, we see that $F$ is an orientable
surface covered by the torus. The cover is necessarily a finite cover by
compactness, hence $F$ itself is the torus, and $\pi_{1}(F)=\mathbb{Z}^{2}$.
This, however, contradicts the assumption that $U\cap\pi_{1}(F)$ is not
finitely generated, as $\mathbb{Z}^{2}$ has no non-finitely generated
subgroups, which finishes the proof.

Finally, we note that the proof that $U$ is commensurable with $\pi_{1}(F)$,
for a certain virual fibering of $M$, is also applicable to the case when $M$
is closed. This addresses the lack of proof of this fact in [9]. ∎

##  3\. Irreducibility of the summands in an irreducible torus sum

An idea which is a key ingredient in the following lemma is due to Peter
Scott.

######  Lemma 3.1.

Let $M=M_{1}\cup_{\mathcal{T}}M_{2}$, or $M=M_{1}\cup_{\mathcal{T}}$, where
$\mathcal{T}\subset M$ is an incompressible torus. If $M$ is irreducible, then
any 2-sphere $S\subset M_{i}-\mathcal{T}$ bounds a 3-ball in
$M_{i}-\mathcal{T}$.

###### Proof.

By abuse of notation, let $S$ be a 2-sphere embedded in $M_{i}-\mathcal{T}$,
for $i=1$ or $2$, and $B$ the 3-ball that it bounds in $M$. We need to show
that $B$ is contained in $M_{i}-\mathcal{T}$. Consider $B^{0}=B-\partial{B}$,
and consider also $B^{0}\cap\mathcal{T}$. Clearly, $B^{0}\cap\mathcal{T}$ is
open in $\mathcal{T}$ as $B^{0}$ is open in $M$. On the other hand, since $S$
and $\mathcal{T}$ are compact, $\inf\left\\{d(x,y):x\in
S,y\in\mathcal{T}\right\\}>0$, for any length metric on $M$ induced by a
Riemannian metric. Therefore, if $\left\\{p_{i}\right\\}$ is a sequence in
$B^{0}\cap\mathcal{T}$ such that $p_{i}\rightarrow p$, we must have $p\in B$
as $B$ is sequentially compact, but $p\notin S$, hence $p\in B^{0}$;
$\mathcal{T}$ is also compact, therefore sequentially compact, hence
$p\in\mathcal{T}$, therefore $p\in B^{0}\cap\mathcal{T}$. Thus, we see that
$B^{0}\cap\mathcal{T}$ is also closed. Because $\mathcal{T}$ is connected,
$B^{0}\cap\mathcal{T}$ is either empty or equals $\mathcal{T}$. We cannot have
$B^{0}\cap\mathcal{T}=\mathcal{T}$ because then $\mathcal{T}\subset B^{0}$,
which would imply that $\mathcal{T}$ is contractible in $M$, thus
contradicting the $\pi_{1}$-injectivity of $\mathcal{T}$. Thus, we have
$B^{0}\cap\mathcal{T}=\emptyset$, hence $B$ is contained in a connected
component of $M-\mathcal{T}$. Since $\partial B\subset M_{i}$, we must also
have $B\subset M_{i}-\mathcal{T}$, thus proving the claim. The case
$M_{1}\cup_{\mathcal{T}}$ is identical - one only needs to consider both tori.
∎

######  Proposition 3.2.

Let $M=M_{1}\cup_{\mathcal{T}}M_{2}$, or $M=M_{1}\cup_{\mathcal{T}}$, where
$\mathcal{T}\subset M$ is an incompressible torus. If $M$ is irreducible, then
$M_{i}$ is irreducible.

###### Proof.

Let $f:S\rightarrow M_{i}$ be an embedding of the 2-sphere into $M_{i}$, for
$i=1$ or $2$. If $f(S)\cap\mathcal{T}=\emptyset$, Lemma 3.1 shows that $f(S)$
bounds a 3-ball in $M_{i}-\mathcal{T}$, hence also in $M_{i}$ thus proving the
claim.

Suppose, therefore, that $f(S)\cap\mathcal{T}\neq\emptyset$. Let $N$ be a
collared neighborhood of $\mathcal{T}\subset\partial M_{i}$ homeomorphic to
$\mathcal{T}\times\left[0,1\right]$, and let $\mathcal{T}$ be identified with
$\mathcal{T}\times\left\\{1\right\\}\subset N$ under the homeomorphism. Will
show that $f(S)$ bounds a 3-ball by first homotoping $f(S)$ to be disjoint
from $\mathcal{T}\subset\partial M_{i}$, and then using the previous result to
show that $f(S)$ is inessential. To this end, let $F$ be the map
$F((x,t),s)=(x,t(1-s/2))$, for $(x,t)\in N$, with $x\in\mathcal{T}$ and
$t\in\left[0,1\right]$, so that $F:N\times I\rightarrow N$. Note that $F$
restricts to the identity on $\mathcal{T}\times\left\\{0\right\\}\subset
M_{i}^{0}$, hence extends to a continuous homotopy
$M_{i}\times\left[0,1\right]\rightarrow M_{i}$ defined to be the identity off
$N$, which we will still denote by $F$, and note also that for this extension
we have $F(\cdot,0)=id_{M_{i}}(\cdot)$. Note, further, that
$F(f(\cdot),0)=f(\cdot)$, and if we set $F(f(\cdot),1)=f_{1}(\cdot)$, then
$f_{1}(p)=(x(p),t(p)/2)$ where $f(p)=(x(p),t(p))$ whenever $f(p)\in N$. It is
obvious that $f_{1}$ is still an embedding of $S$ into $M_{i}$ since $f_{1}$
is injective, as clearly $f_{1}(p_{1})=f_{1}(p_{2})$ is impossible in either
of the two cases: $f_{1}(p_{1}),f_{1}(p_{2})\in N$ or
$f_{1}(p_{1}),f_{1}(p_{2})\notin N$. We also have
$f_{1}(S)\cap\mathcal{T}=\emptyset$. By Lemma 3.1, $f_{1}(S)$ bounds a 3-ball.
Let $\phi:B\rightarrow M_{i}$ be this embedding of the 3-ball into $M_{i}$.
Since $\phi|_{\partial B}(p)=f_{1}(p)=F(f(p),1)$, for $p\in S=\partial B$, the
map $\phi:B\rightarrow M_{i}$ and the map $S\times\left[0,1\right]\rightarrow
M_{i}$, defined by $(p,t)\rightarrow F(f(p)),t)$, fit together to give a
continuous map from $\overline{\phi}:B\cup_{\partial B\cong
S\times\left\\{1\right\\}}\left(S\times\left[0,1\right]\right)\cong
B\rightarrow M_{i}$. Thus, we see that the emedded 2-sphere
$f(S)=\overline{\phi}(S\times\left\\{0\right\\})$ is inessential in $M_{i}$.
This happens if and only if $f(S)$ bounds a compact simply connected
submanifold $W$ of $M_{i}$, see Lemma 9.1.8 in [12]. Now, since
$H_{1}(W)=\pi_{1,ab}(W)$, we see that $H_{1}(W)=0$ and therefore $f(S)$ is
null homologous in $W$. By considering the 3-cycles in $W$ whose boundary is
precisely $f(S)$, we see that each 3-simplex of $W$ must be included in such a
3-cycle, as we cannot have any free faces not in $f(S)$, and we further
observe that $f(S)$ is the only boundary component of $W$. Now, after
attaching a 3-ball to $W$ along the boundary sphere and applying Van Kampen’s
Theorem, we see that we have a simply connected closed 3-manifold, which must
be $\mathbb{S}^{3}$ by the positive solution to the Poincaré Conjecture. Thus,
we conclude that $W$ is itself a 3-ball, which concludes the proof. ∎

We record here an immediate corollary to Theorem 5.3 in [10], as corrected in
[11]:

##  4\. An application to fiberings of compact 3-manifolds over
$\mathbb{S}^{1}$

As an application of Theorem 2.1, I give the following result which
generalizes Theorem 2.10 in [9], and which is now vindicated as a true
statement of mathematical fact, see [11]. The reader should know that this
generalization was only possible after the proof of the Geometrization Theorem
by Perelman in 2003. In light of Proposition 3.2, this result, which first
appeared in [10] and [11], now becomes:

######  Theorem 4.1.

Let $M$ be a closed 3-manifold. If $G=\pi_{1}(M)$ contains a finitely
generated subgroup $U$ of infinite index in $G$ which contains a nontrivial,
subnormal subgroup $N$ of $G$, then: (a) $M$ is irreducible, (b) if further:

  1. (1)

$N$ has a subnormal series of length $n$ in which $n-1$ terms are assumed to
be finitely generated, and

  2. (2)

either all inclusions $N_{i}\hookrightarrow N_{i+1}$, for $i>0$, are of finite
index, or there exist (at least two) indices $i_{0}$ and $i_{1}$, $i_{0}\neq
i_{1}$, $i_{0},i_{1}>1$, such that the inclusions $N_{i}\hookrightarrow
N_{i+1}$ are of infinite index for $i=i_{0},i_{1}$, or $N=N_{0}$ is finitely
generated and there exists (at least) one value of the index $i$ for which the
inclusion $N_{i}\hookrightarrow N_{i+1}$ is of infinite index, and

  3. (3)

$N$ intersects nontrivially the fundamental groups of the splitting tori of
some decomposition $\mathfrak{D}$ of $M$ into geometric pieces, and
$N\cap\pi_{1}(X_{i})\neq\mathbb{Z}$ for all geometric pieces
$X_{i}\in\mathfrak{D}$,

then, $M$ has a finite cover which is a bundle over $\mathbb{S}$ with fiber a
compact surface $F$ such that $\pi_{1}(F)$ and $U$ are commensurable. Further,
any decomposition of $M$ along a collection of incompressible tori yields
pieces which are irreducible, in particular, each $X_{i}\in\mathfrak{D}$ is
irreducible.

###### Proof.

The proof of the theorem is essentially contained in [10] and [11]. The
corrigendum [11] notes that, as of the time of its publication, the theorem
was contingent on the proof of a result at least as general as Theorem 2.2 for
compact manifolds with nonempty boundary, due to the absence of such a proof
from [9], which I had somehow overlooked at the time. Therefore, the only
statement which needs proof is the claim about the irreducibility of the
$X_{i}$; this, however, follows immediately from Proposition 3.2 and
induction. ∎

An easier to state special case of the preceding result is the following:

######  Corollary 4.2.

Let $M$ be a compact 3-manifold with empty or toroidal boundary. If
$G=\pi_{1}(M)$ contains a finitely generated subgroup $U$ of infinite index in
$G$ which contains a nontrivial, normal subgroup $N$ of $G$, then: (a) $M$ is
irreducible, (b) if further:

  1. (1)

$N$ intersects nontrivially the fundamental groups of the splitting tori of
some decomposition $\mathfrak{D}$ of $M$ into geometric pieces, and

  2. (2)

$N\cap\pi_{1}(X_{i})\neq\mathbb{Z}$ for all geometric pieces
$X_{i}\in\mathfrak{D}$,

then, $M$ has a finite cover which is a bundle over $\mathbb{S}$ with fiber a
compact surface $F$ such that $\pi_{1}(F)$ and $U$ are commensurable, and each
$X_{i}$ is an irreducible, compact, geometric 3-manifold.

##  5\. Acknowledgements

I would like to posthumously express my sincerest gratitude to the late Prof.
Peter Scott for the many years of patient advice and his friendship, and to
thank his son David for allowing me, on behalf of Prof. Scott’s estate, to use
his unpublished notes on 3-manifolds. I would also like to acknowledge and
express my heartfelt thanks for the very timely and very helpful advice Chris
Mooney gave me during that brief and fateful conversation we had back in 2011,
which resulted in Lemmas 3.1 and 3.2 in my very first paper published in the
Journal of Groups, Complexity, Cryptography in 2015, and to Enric Ventura, who
brought to my attention the immediate converse of the main result, see
Proposition 3.7, in this older article. I am also very thankful to Moon Duchin
for introducing me to Chris Mooney and for giving my first paper’s initial
draft a first read through at a time when nothing seemed to be going according
to plan for me: stuck on two thesis problems, with my time to finish my degree
nearly having expired. I would also like to thank the Elmhurst Public Library
of the City of Elmhurst, Illinois for allowing me to use its quiet area to
read and work in, and the Founders Memorial Library of Northern Illinois
University for extending me courtesy borrowing privileges, which have been
invaluable. Finally, I would like to thank all the mathematicians who post
useful bits of mathematical advice and references on mathoverflow.net.

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# Quantitative analysis of the interaction between a dc SQUID and an
integrated micromechanical doubly clamped cantilever

Majdi Salman  Electronic mail<EMAIL_ADDRESS>Georgina M Klemencic
Soumen Mandal  Scott Manifold  School of Physics and Astronomy, Cardiff
University, Queen’s Building, The Parade,
Cardiff, CF24 3AA, United Kingdom  Luqman Mustafa  Centre for Innovation
Competence SiLi-nano, Martin-Luther-University Halle-Wittenberg, Karl-
Freiherr-von-Fritsch-Strasse 3, 06120 Halle (Saale), Germany  Oliver A
Williams  Sean R Giblin  School of Physics and Astronomy, Cardiff University,
Queen’s Building, The Parade,
Cardiff, CF24 3AA, United Kingdom

###### Abstract

Based on the superconducting quantum interference device (SQUID) equations
described by the resistively- and capacitively-shunted junction model coupled
to the equation of motion of a damped harmonic oscillator, we provide
simulations to quantitatively describe the interaction between a dc SQUID and
an integrated doubly clamped cantilever. We have chosen to investigate an
existing experimental configuration and have explored the motion of the
cantilever and the reaction of the SQUID as a function of the voltage-flux
$V(\Phi)$ characteristics. We clearly observe the Lorentz force back-action
interaction and demonstrate how a sharp transition state drives the system
into a nonlinear-like regime, and modulates the cantilever displacement
amplitude, simply by tuning the SQUID parameters.

###### pacs:

Valid PACS appear here

††preprint: AIP/123-QED

##  I  Introduction

Theoretical and experimental studiesEtaki ; Buks ; Armour ; Fazio ; DArmour ;
Ventra ; Ella  of linear and nonlinear micro and nanomechanical resonators are
of great interest as they can be used for sensitive force and displacement
measurements. The physical parameters of the resonators can also be tuned to
observe the transition from the classical to quantum regimes with relative
experimental ease, enabling observations of macroscopic quantum
systems.Schwab1  Significant experimental progress in the detection of
resonators as they enter the quantum ground state has been achieved by
capactiave coupling to superconducting flux qubits,Connell  and quantum state
control of a mechanical drum resonator in a superconducting resonant circuit
has been achieved by phonon-photon coupling.Teufel ; CSchwab  The state
detection is an integral part of any coupled resonator system as the coupling
mechanism is implicit in any experimental endeavour.

Considering a doubly clamped cantilever, it is obvious that as the cantilever
oscillates the displacement changes, and the transduction technique will cause
a back-action that influences the cantilever position.Caves  The impact of
back-action can be positive in terms of coolingDArmour  and squeezing the
resonator motion,Almog ; Liu ; Zoller  and coupling and synchronising multiple
resonators.Wiesenfeld ; Rogers  Depending on the specific transduction
technique, back-action can be due to radiation pressure,Kippenberg  electron
tunnelling,Steele  or photothermal effects.Jourdan

Figure 1: (a) scheme for the dc SQUID displacement detector in which the two
Josephson junctions are labelled by $\mathrm{J_{1}}$ and $\mathrm{J_{2}}$. The
cantilever displacement is out-of-plane, and the applied magnetic field, $B$,
is in-plane. (b) $V(\Phi)$ characteristics for a dc SQUID with
$\beta_{L}=0.115$ and $\beta_{C}=1.61$. Four regimes are identified: $(i)$ the
simple oscillatory regime where the bias current, $I_{b}=2.0I_{0}$. The other
regimes are: $(ii)$ the rapidly changing regime (red), $(iii)$ the zero
voltage response regime (blue), and $(iv)$ the intermediate regime (green).

Previous experiments have used a dc SQUID to detect the motion of a suspended
doubly clamped cantilever integrated directly into a SQUID loop,Etaki ; Poot
and for a torsional SQUID cantilever.SEtaki  For this SQUID-based transduction
scheme, the back-action has a simple inductive component caused by the Lorentz
force due to the circulating current.SEtaki ; Poot ; Shevchuk  Experimentally,
the Lorentz back-action was shown to shift the mechanical cantilever resonant
frequency and quality factor by $\Delta f$ and $\Delta Q$ respectively. To
understand the effect of back-action on $\Delta f$ and $\Delta Q$, two
transfer functions were obtained,Poot  which are coefficients for the average
circulating current expanded in the terms of the cantilever displacement, $u$,
and velocity, $\dot{u}$.

In previous work, however, it was not possible to obtain the velocity-
dependent transfer function in the frame of the SQUID equations coupled to the
equation of motion of the doubly clamped cantilever. To simplify this issue,
Poot et alPoot  modulated the flux change in the SQUID loop caused by the
cantilever oscillation. Subsequently, the total flux in the SQUID loop was
assumed to be a function of the externally applied flux,
$\Phi_{\mathrm{ext}}$, and the modulation of the flux due to the changing area
of the loop,
$\Phi\rightarrow\Phi_{\mathrm{ext}}+\Phi_{\mathrm{mod}}\cos(\omega_{\mathrm{mod}}t)$.
Such a modulation can describe the influence of the back-action on $\Delta f$
and $\Delta Q$ of the cantilever when the SQUID displacement detector is tuned
within limited regions of the $V(\Phi)$ curve. Poot  However, a full
description of the SQUID-cantilever interaction requires a comprehensive model
to provide information not only about the influence of back-action in all
regions of $V(\Phi)$, but also about the amplitude, width, line shape, and
responsivity, $\frac{\mathrm{d}V}{\mathrm{d}u}$, which must be calculated by
linking the cantilever displacement to the SQUID voltage. Thus, the need for
quantitative treatments of the unscaled SQUID equations coupled explicitly to
the equation of motion for the integrated beam becomes important. Though such
treatments are complicated and challenging,SupEtaki  they can be performed
numerically with improving computational capabilities.

In this paper, we simulate the interaction between a dc SQUID and an embedded
micromechanical doubly clamped cantilever as experimentally demonstrated by
Etaki et al Etaki  and shown schematically in Fig. 1(a). The SQUID-cantilever
interaction is analysed in different regions of the $V(\Phi)$ curve, as shown
in Fig. 1(b). Within this framework, we have explored some regions of the
$V(\Phi)$ curve, where the SQUID-cantilever response is apparently strongly
nonlinear. Futhermore, the back-action and the subsequent response of the
SQUID is linked to the cantilever displacement. The effect of changing the
SQUID operating point is discussed in depth, and it is demonstrated that the
SQUID itself can be used to control the cantilever response by simple
modification of the controllable SQUID parameters.

##  II  The Model

The model we present is based on the experimental parameters of Etaki et al
Etaki  to allow for experimental verification of the results. As such the
inductive screening parameter, $\beta_{L}$, and Stewart-McCumber parameter,
$\beta_{C}$, are selected to be $0.115$ and $1.61$ respectively. With these
values for $\beta_{L}$ and $\beta_{C}$, $V(\Phi)$ characteristics of an
overdamped dc SQUID are shown in Fig. 1(b) to demonstrate the possible
operating points of a SQUID displacement detector. The $V(\Phi)$ curves are
calculated using the time-scaled SQUID equations described by the resistively-
and capacitively-shunted junction (RCSJ) model Clarke . In Fig. 1(b), four
different regimes in the SQUID $V(\Phi)$ response are defined: $(i)$ the
simple oscillatory regime where the bias current, $I_{b}=2.0I_{0}$. The other
regimes are $(ii)$ the rapidly changing regime (red), $(iii)$ the zero voltage
response regime (blue), and $(iv)$ the intermediate regime (green). Our
analysis covers the interaction between a dc SQUID and an integrated
cantilever when the system is tuned to operating points within these defined
regimes, and the resulting effect on the cantilever-SQUID dynamics.

Figure 2: Line shapes for (a) SQUID voltage and (c) cantilever displacement
calculated as a function of $\Phi_{\mathrm{ext}}$ for $I_{b}=2.0I_{0}$. (b)
and (d) density plots for SQUID voltages and cantilever displacement
respectively. (e) the linear displacement-voltage trace as extracted by
linking (b) and (d) via the frequency. (f) the responsivity
($\frac{\mathrm{d}V}{\mathrm{d}u}$) as calculated from the slopes of the
displacement-voltage lines.

We use the equation of motion of a damped harmonic oscillator given in Poot
to describe the displacement, $u(t)$, of the mechanical cantilever:

$m\ddot{u}+\frac{m\omega_{0}}{Q_{0}}\dot{u}+m\omega^{2}_{0}u=F_{d}(t)+F_{L}(t),$
(1)

where $m$ is the beam mass, $\omega_{0}=2\pi f_{0}$ is the intrinsic
frequency, $Q_{0}$ is the quality factor, $F_{d}=F_{0}\cos(\omega_{0}t)$ is
the driving force, and $F_{L}(t)$ is the Lorentz force
$F_{L}(t)=aB\ell(I_{b}/2+J)$. Here, $B$ is the in-plane magnetic field, $\ell$
is the length of the cantilever, $J$ is the circulating current, and $a=0.91$
Etaki  is a geometrical factor that depends on the mode shape. Eq. (1) is
coupled to the dc SQUID equations given by the RCSJ model:

$\frac{\Phi_{0}}{2\pi}C\ddot{\delta_{1}}+\frac{\Phi_{0}}{2\pi}C\frac{1}{R}\dot{\delta_{1}}+I_{0}\sin(\delta_{1})=\frac{1}{2}(I_{b}+J),$
(2)

$\frac{\Phi_{0}}{2\pi}C\ddot{\delta_{2}}+\frac{\Phi_{0}}{2\pi}C\frac{1}{R}\dot{\delta_{2}}+I_{0}\sin(\delta_{2})=\frac{1}{2}(I_{b}-J),$
(3)

$\delta_{1}-\delta_{2}=2\pi\cdot{\Phi_{\mathrm{tot}}}/{\Phi_{0}},$ (4)

where $\delta_{1,2}$ are the phase differences of the junctions, $\Phi_{0}$ is
the flux quantum, $I_{b}$ is the bias current, $I_{0}$ is the critical
current. The total flux, $\Phi_{\mathrm{tot}}$, has three contributions: $(i)$
the external flux $\Phi_{\mathrm{ext}}$, $(ii)$ the flux due to the
circulating current, $J$, flowing through the inductance of the loop, $L$, and
$(iii)$ the change in flux through the loop due to the cantilever
displacement, $aB\ell u$. Therefore,
$\Phi_{\mathrm{tot}}=\Phi_{\mathrm{ext}}+LJ+aB\ell u(t)$, and Eqs. (1-3) are
coupled via the circulating current as
$J=\frac{1}{L}(\frac{\delta_{1}-\delta_{2}}{2\pi}\Phi_{0}-\Phi_{\mathrm{ext}}-aB\ell
u)$.

These coupled differential equations are numerically solved without averaging
the SQUID voltage and circulating currents, or scaling the time. Therefore,
the time span $T_{\mathrm{max}}$ must be large enough to be suitable for the
cantilever, while the time step $\mathrm{d}t$ must be small enough to resolve
the impact of the fast changes dominated by the relatively high SQUID
characteristic frequency $\omega_{c}=\frac{2\pi RI_{0}}{\Phi_{0}}$. Although
this can be computationally expensive for cantilevers with very low
frequencies relative to $\omega_{c}$, the experimental results of Etaki et al
Etaki  allow their experiment to be modelled within a relatively small time
window.

Here, we solve a system for identical experimental conditions demonstrated by
Etaki et al Etaki  with $f_{0}\simeq 2$ MHz. To calculate the time dependent
voltage, $V=\Phi_{0}\frac{\dot{\delta_{1}}+\dot{\delta_{2}}}{2\pi}$, the
Runge-Kutta method (RK4) was used to numerically integrate the equations
presented above. The SQUID response was then obtained in the frequency domain
by evaluating the Fourier transform of the SQUID voltage and the cantilever
displacement. Our calculations were performed for $I_{0}=0.7$ $\mu$A, $R=29.5$
$\Omega$, $B=111$ mT, $C=0.91$ pF, and $L=170$ pH. These values give a
McCumber-Stewart parameter $\beta_{C}=\frac{2\pi I_{0}R^{2}C}{\Phi_{0}}=$ 1.61
and a screening parameter $\beta_{L}=\frac{2I_{0}L}{\Phi_{0}}=0.115$. The
cantilever has a length $\ell=50$ $\mu$m, a mass $m=6\times 10^{-13}$ kg, and
was assumed to have a resonant frequency $f_{0}=2.0018$ MHz and a quality
factor $Q_{0}=25000$. The piezo drive which controls $F_{\mathrm{d}}$ is used
only to locate the eigenmodes and is turned off during measurement SEtaki .
Thus, at $t=0$ the initial velocity
$v_{0}=\left.\frac{\mathrm{d}u}{\mathrm{d}t}\right|_{u=u_{0}}=0$, where
$u_{0}$ is the initial displacement amplitude. Here, $u_{0}=20$ pm.

The time span chosen for these calculations was $T_{\mathrm{max}}=25$ ms, i.e.
more than six times the lifetime of the cantilever, and the optimised time
step chosen was $\mathrm{d}t=0.0125$ ns. The calculations were repeated at
different values of normalised flux in the range
$0.90\Phi_{0}\leq\Phi_{\mathrm{ext}}\leq 0.05\Phi_{0}$, and bias currents in
the range $2.0I_{0}\leq I_{\mathrm{b}}\leq 1.10I_{0}$. In the frequency
domain, we selected frequency steps of $\mathrm{d}f=12.5$ Hz. The units of the
response which were calculated directly from a Fourier transform are
V$\mathrm{\cdot}$s for the unnormalised SQUID voltage and $\mathrm{m.s}$ for
the unnormalised cantilever displacement. To convert the units of the voltage-
response from V$\mathrm{\cdot}$s to $\mathrm{V}$, the response was multiplied
by $\frac{1}{\tau}$, where $\tau$ is the lifetime of the cantilever, which is
related to the full width at half maximum (FWHM) as
$\frac{1}{\pi\tau}=f_{\mathrm{FWHM}}$. A similar procedure was used to convert
the units of the displacement-response from m$\mathrm{\cdot}$s to
$\mathrm{m}$.

##  III Results

###  III.1  The simple oscillatory regime behaviour

To ensure our calculations are based in physical reality, we contextualized
the calculations with the existing experimental parameters.Etaki  As
experimentally demonstrated, the voltage responses exhibit Lorentzian
distributions and for $\Phi_{\mathrm{ext}}=0.25\Phi_{0}$, i.e. the highest
SQUID sensitivity for $I_{b}=2.0I_{0}$ shown in Fig. 1(b), there was no
relative experimental shift in $\Delta f$ of the cantilever. Changing the
operating point of the SQUID by changing $\Phi_{\mathrm{ext}}$ within the
simple oscillatory region shown in Fig. 1(b) affects $\Delta f$, and the
operating point clearly affects the sensitivity to the SQUID voltage as
clearly shown in Fig. 2(a) and (b). Moreover the subsequent cantilever
displacement is also affected (Fig. 2(c) and (d)). These results clearly
demonstrate the influence of the Lorentz back-action on the resonator from the
SQUID displacement detector, and the expected magnitude of change in the
experimental variables.

Figure 3: Calculations for the range $0.75\Phi_{0}\leq\Phi_{\mathrm{ext}}\leq
0.25\Phi_{0}$ and $\mathrm{2.0I_{0}}\leq I_{\mathrm{b}}\leq\mathrm{1.55I_{0}}$
for (a) the frequency shift, $\Delta f$, (c) FWHM and (e) the maximum SQUID
voltage, $V_{\mathrm{max}}$. The corresponding density plots are shown in (b),
(d), and (f) respectively.

The cantilever displacement and SQUID voltage are explicitly linked in the
frequency domain, i.e., the displacement $u(f)$ is parametrically linked to
the voltage $V(f)$. The subsequent analysis was performed at $I_{b}=2.0I_{0}$
and $0.05\Phi_{0}\leq\Phi_{\mathrm{ext}}\leq 0.95\Phi_{0}$, and the
displacement-voltage trace is plotted in Fig. 2(e). The traces show a linear
dependence of voltage on displacement, which allows determination of the
cantilever position in a responsivity specified by the slope of the
displacement-voltage lines. Consequently, the responsivity
($\frac{\mathrm{d}V}{\mathrm{d}u}$) was calculated at $I_{b}=2.0I_{0}$ for
different $\Phi_{\mathrm{ext}}$ values, with the result shown in Fig. 2(f).
The figure shows a sinusoidal behaviour for $\frac{\mathrm{d}V}{\mathrm{d}u}$
which varies from $4.7\times 10^{-2}~{}\mathrm{nV.fm}^{-1}$ at
$\Phi_{\mathrm{ext}}=0.25\Phi_{0}$ to $0.5\times
10^{-2}~{}\mathrm{nV.fm}^{-1}$ at $\Phi_{\mathrm{ext}}=0.50\Phi_{0}$.
Importantly, Fig. 2 shows an appropriate representation of the experimental
results by Etaki et al Etaki , thereby demonstrating a good computational
model.

###  III.2  The intermediate regime behaviour

Further calculations were performed through the $V(\Phi)$ curve identified in
Fig. 1(b) to examine the SQUID-cantilever coupling and explore the system
response as the coupling/back-action is modified. Fig. 3(a)-(f) shows $\Delta
f$, the FWHM, and the SQUID voltage as the bias current and
$\Phi_{\mathrm{ext}}$ are tuned. The largest frequency shift corresponds to
the smallest gradient ($\frac{\mathrm{d}V}{\mathrm{d}\Phi}$) of the working
point. This can clearly be understood by Eq. (1), where the frequency of the
cantilever is controlled by the displacement coefficient. As the cantilever
frequency is shifted by changing $\Phi_{\mathrm{ext}}$ and $I_{b}$, a
modification in this coefficient emerges due to the circulating current
dependence on $u$. Such a dependence was previously analysed by expanding the
circulating current in terms of the displacement, $u$ Poot . In this way, the
new displacement coefficient, which arises from the back-action of the SQUID
current on the cantilever, modifies the frequency and causes a slight or
significant shift depending on $\Phi_{\mathrm{ext}}$ and $I_{b}$.

The Lorentz back-action also affects the cantilever quality factor; FWHMs of
simulated line shapes are extracted and presented as a function of
$\Phi_{\mathrm{ext}}$ for various values of $I_{b}$ in Fig. 3(c) and (d). The
variation of the FWHM can be interpreted in an identical way to that of
$\Delta f$, where the only difference being that
$\mathrm{FWHM}=\frac{\omega_{0}}{2\pi Q_{0}}$ is given in terms of velocity
coefficient in Eq. 1. Thus, the FWHM is modified if $J$ is assumed to have a
dependence on the velocity in addition to the displacement which modifies the
frequency Poot . The corresponding peak voltage, $V_{\mathrm{max}}$,
dependence on $\Phi_{\mathrm{ext}}$ and $I_{b}$ is shown in Fig. 3(e) and (f).
The behaviour of $V_{\mathrm{max}}$ as a function of $\Phi_{\mathrm{ext}}$ is
consistent with $\mathrm{d}V/\mathrm{d}\Phi_{\mathrm{ext}}$ of the SQUID
$V(\Phi_{\mathrm{ext}})$ curve shown in Fig. 1(b).

Figure 4: (a) unnormalised displacement and (b) corresponding unnormalised
SQUID voltage when the SQUID displacement detector is tuned
($\Phi_{\mathrm{ext}}=0.30\Phi_{0}$ and $I_{b}=1.20I_{0}$) to the working
point shown in the inset of (a). The initial cantilever amplitudes are
$u_{0}=20~{}\mathrm{pm}$ (blue), $u_{0}=10~{}\mathrm{pm}$ (red), and
$u_{0}=5~{}\mathrm{pm}$ (black), which correspond to a change of flux in the
SQUID loop of $0.05\Phi_{0}$, $0.025\Phi_{0}$, and $0.0125\Phi_{0}$
respectively.

Figure 5:  (a) Snapshot for the time array of cantilever displacements at (a)
$\Phi_{\mathrm{ext}}=0.25\Phi_{0}$ and $I_{b}=2.0I_{0}$ (a point in the simple
oscillatory regime) versus (b) a point $\Phi_{\mathrm{ext}}=0.30\Phi_{0}$ and
$I_{b}=1.20I_{0}$ in the rapidly changing regime in which a sharp transition
state emerges until the cantilever enters the normal state at
$u=u_{\mathrm{N}}$. (c) calculations for the normal state positions of
specific lines around $\Phi_{\mathrm{ext}}=0.30\Phi_{0}$ and
$I_{b}=1.20I_{0}$, and (d) the yellow-blue islands in the density plot
indicate a shift in the normal state positions that starts emerging at
$t=t_{\mathrm{N}}$ and $u=u_{\mathrm{N}}$.

###  III.3  The rapidly changing regime behaviour

Now we turn to a different regime from Fig 1(b), where the largest effect of
back-action on the cantilever is observed, and the SQUID response is
apparently nonlinear. To examine the effect of back-action on the cantilever
motion, a point in such region was selected as shown in the inset of Fig.
4(a). Subsequently, at $\Phi_{\mathrm{ext}}=0.30\Phi_{0}$ and
$I_{b}=1.20I_{0}$, the unnormalised cantilever displacement and corresponding
unnormalised SQUID voltage response for various displacement amplitudes
$u_{0}$, are obtained and plotted in Fig. 4(a) and Fig. 4(b). When
$u_{0}=20~{}\mathrm{pm}$, the cantilever appears to have a nonlinear behaviour
as demonstrated by the modified line shape of the cantilever and the SQUID
response. Specifically, as the displacement is reduced from
$u_{0}=20~{}\mathrm{pm}$ to $u_{0}=10~{}\mathrm{pm}$ to $u_{0}=5\mathrm{pm}$,
the change in the flux through the loop is $0.05$, $0.025$ and
$0.0125\Phi_{0}$ respectively. This change of flux in the SQUID loops drives
the cantilever to experience two $V(\Phi)$ regions of different responsivity,
which results in the nonlinear-like behaviour. It should be noted, however,
that as the cantilever returns to its dynamical equilibrium position, the
response becomes more Lorentzian as expected.

The effect of the SQUID-cantilever interaction on the cantilever motion can be
more clearly observed by comparing the time evolution of the cantilever
displacement for two different bias and flux values. The time dependent
displacement for $\Phi_{\mathrm{ext}}=0.25\Phi_{0}$ and $I_{b}=2.0I_{0}$ is
plotted in Fig. 5(a), and for $\Phi_{\mathrm{ext}}=0.30\Phi_{0}$ and
$I_{b}=1.20I_{0}$ in Fig. 5(b). Clearly if the SQUID operating point is in the
rapidly changing regime (Fig. 5b) there is a sharp transition state as the
cantilever returns to its equilibrium position. Naively if the SQUID bias is
switched when the cantilever motion is large, there is an instantaneous
damping which can be used to modify the motion of the cantilever. Normal state
positions, $u_{\mathrm{N}}$, for specific lines around
$\Phi_{\mathrm{ext}}=0.30\Phi_{0}$ and $I_{b}=1.20I_{0}$ are shown in Fig.
5(c). These positions are extracted when the cantilever enters the normal
state that accounts for the Lorentzian profile in the frequency domain, and
when the amplitude starts decaying exponentially at time $t=t_{\mathrm{N}}$,
as shown in Fig. 5(b). A more comprehensive analysis is presented in the
density plot shown in Fig. 5(d). The plot given in Fig. 5(c) exhibits details
for one of the yellow-blue islands in the density plot. The islands correspond
to the intermediate regimes in the $V(\Phi)$ curves. Its anticipated that such
effect could be employed to precisely and rapidly control the amplitude of the
cantilever displacement below its initial amplitude which can be set by a
piezo drive used to locate the eigenmodes of the cantilever. In other words,
putting the system in such regions enables modulating the cantilever amplitude
after isolating the system from the external actuator.

##  IV Conclusion

In conclusion, we have shown how the tuning of the SQUID device affects the
back-action between the SQUID and the doubly clamped cantilever. Specifically,
we have quantified the line shapes expected from the SQUID response and the
corresponding cantilever displacement. The effect can be quantitatively
analysed via the shift in the cantilever frequency, the line width, intensity,
and shift in the position of the normal state. Direct solutions for the
unscaled dc SQUID equations coupled to the equations of motion of an
integrated cantilever allow determination of voltage-displacement traces of a
displacement detector. For a SQUID displacement detector tuned to a working
point in the rapidly changing region, a sharp transition state emerges and a
nonlinear-like response due to the emergence of such state is observed. This
state could be used to employ the system as a self modulator for the
displacement amplitude of the cantilever. These results should allow a clearer
understanding and manipulation of future experimental work.

###### Acknowledgements.

The authors would like to acknowledge Andrew Armour for numerous conversations
and beneficial comments and N. Peretto, R. M. Smith, and R. A. Frewin for the
computational facilities, and E. Riordan for reading and discussions. We also
gratefully acknowledge support by the European Research Council under the EU
Consolidator Grant ”SUPERNEMS” (Project No. 647471).

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