# Finite rigid sets and the separating curve complex

Junzhi Huang and Bena Tshishiku

###### Abstract

For each closed surface $\Sigma$ of genus $g\geq 3$, we find a finite
subcomplex of the separating curve complex that is rigid with respect to
incidence-preserving maps.

##  1 Introduction

This paper is about rigidity properties of the complex
$\mathcal{C}^{s}(\Sigma)$ of separating curves on a closed, oriented surface
$\Sigma$ of genus $g\geq 3$.

For a subcomplex $Z\subset\mathcal{C}^{s}(\Sigma)$, a simplicial map
$\phi:Z\rightarrow\mathcal{C}^{s}(\Sigma)$ is said to be _incidence
preserving_ if $i(\alpha,\beta)\neq 0$ implies that
$i(\phi(\alpha),\phi(\beta))\neq 0$ for vertices $\alpha,\beta$ of $Z$. Here
$i(\cdot,\cdot)$ denotes the geometric intersection number (see §2 for
definitions).

The goal of this paper is to prove the following result.

######  Theorem A (Finite rigid set of separating curves).

Fix a closed, oriented surface $\Sigma$ of genus $g\geq 3$. There is a finite
subcomplex $Y^{s}\subset\mathcal{C}^{s}(\Sigma)$ such that any incidence-
preserving map $\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$ is induced by an
extended mapping class, i.e. there exists
$h\in\operatorname{Mod}^{\pm}(\Sigma)$ such that $h\bigr{|}_{Y^{s}}=\phi$.
Furthermore, $h$ is unique.

The construction of $Y^{s}$ is explicit and is given in §3. We assume $g\geq
3$ because the $\mathcal{C}^{s}(\Sigma_{2})$ is discrete and
$\mathcal{C}^{s}(\Sigma_{1})$ is empty. We also note that the case $g=3$ is
easy because we can take $Y^{s}$ to be a single vertex. Below we restrict to
$g\geq 4$, an assumption necessary for our argument.

######  Remark 1.1 (Incidence preserving vs. superinjective).

Incidence-preserving maps
$\mathcal{C}^{s}(\Sigma)\rightarrow\mathcal{C}^{s}(\Sigma)$ are typically
called _superinjective maps_ in the literature. This terminology originates in
work of Irmak [Irm04]. We do not use the term ``superinjective" because it
would be misleading, since for general $Z\subset\mathcal{C}^{s}(\Sigma)$, an
incidence-preserving map $Z\rightarrow\mathcal{C}^{s}(\Sigma)$ need not be
injective.

######  Remark 1.2 (Similar rigidity results).

Theorem A may be viewed as a ``hybrid" of two results.

  * •

In [AL13], Aramayona–Leininger construct a finite subcomplex $X$ of the curve
complex $\mathcal{C}(\Sigma)$ with the property that any locally injective map
$X\rightarrow\mathcal{C}(\Sigma)$ is induced by an extended mapping class.

  * •

In [BM04], Brendle–Margalit prove that any incidence-preserving map
$\mathcal{C}^{s}(\Sigma)\rightarrow\mathcal{C}^{s}(\Sigma)$ is induced by an
extended mapping class.

These results have as common ancestor Ivanov's rigidity result
$\operatorname{Aut}(\mathcal{C}(\Sigma))\cong\operatorname{Mod}^{\pm}(\Sigma)$
(see [Iva97]), which itself is motivated by the fundamental theorem of
projective geometry.

For other results about finite rigid subcomplexes, see Shinkle's work [Shi20,
Shi21].

One may view Theorem A (and the similar aforementioned rigidity results) as
curve-complex analogues of the fact that an isometry of a Riemannian manifold
is determined by its values on a small open set.

######  Remark 1.3 (Different notions of rigidity).

For the subcomplex $Y^{s}$ constructed in the proof of Theorem A, one can ask
if any _locally injective_ map $Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$ is
induced by an extended mapping class, in analogy to the result of [AL13]. We
do not know whether or not this is true, and it seems difficult to prove using
the same strategy as [AL13].

Theorem A raises two questions:

######  Question 1.4.

Is the finite rigid subset $Y^{s}\subset\mathcal{C}^{s}(\Sigma)$ produced in
Theorem A homologically nontrivial in $\mathcal{C}^{s}(\Sigma)$?

The analogue of Question 1.4 for the Aramayona–Leininger rigid set
$X\subset\mathcal{C}(\Sigma)$ is asked in [AL13, Question 2] and is answered
by Birman–Broaddus–Menasco [BBM15]. One difficulty in answering Question 1.4
is that the homotopy type of $\mathcal{C}^{s}(\Sigma)$ is not completely
known. Looijenga proved that $\mathcal{C}^{s}(\Sigma)$ is $(g-3)$-connected
[Loo13], and Looijenga and van der Kallen [vdKL11] proved that the quotient
$\mathcal{C}^{s}(\Sigma)/\mathcal{I}_{g}$ by the Torelli group is homotopy
equivalent to a wedge of $(g-2)$-dimensional spheres. The complex $Y^{s}$ that
we define has dimension $2g-4$, which is equal to the dimension of
$\mathcal{C}^{s}(\Sigma)$.

In §7, we construct a rigid set in the genus-3 separating curve complex that
is homotopy equivalent to a wedge of 21 circles. This gives a very concrete
example to test Question 1.4.

######  Question 1.5.

Does there exist finite rigid subsets of $\mathcal{C}^{s}(\Sigma)$ of
arbitrarily large diameter?

The analogue of Question 1.5 for $\mathcal{C}(\Sigma)$ is asked in [AL13,
Question 1] and answered affirmatively in [AL16].

###  1.1 Strategy of the proof

To define $Y^{s}$, we first define a subcomplex $Y$ of the (full) curve
complex $\mathcal{C}(\Sigma)$, and then we define
$Y^{s}=Y\cap\mathcal{C}^{s}(\Sigma)$ (see §3). The proof proceeds in the
following steps.

#### Step 1 (rigidity of $Y$).

We show that any locally injective map $Y\rightarrow\mathcal{C}(\Sigma)$ is
induced by a mapping class (Proposition 4.1). To prove this, we use that our
$Y$ contains the finite rigid set $X\subset\mathcal{C}(\Sigma)$ of
Aramayona–Leininger [AL13]. Although rigidity is not generally inherited by
supersets, we can show $Y$ is rigid by showing that each element of
$Y\setminus X$ is _uniquely determined_ (in the sense of [AL13, Defn. 2.1]) by
a subset of $X$.

Given Step 1, we would like to show that any incidence-preserving map
$\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$ extends to a locally injective
map $\widetilde{\phi}:Y\rightarrow\mathcal{C}(\Sigma)$. For then Step 1
implies that $\widetilde{\phi}$, and hence $\phi$, is induced by an extended
mapping class $h$. Furthermore, $h$ is unique by the uniqueness part of [AL13,
Thm. 5.1].

To define an extension, we need to define $\widetilde{\phi}(y)$ for
nonseparating curves. We do this using _sharing pairs_ (defined by
Brendle–Margalit [BM04, §4]), which give a way to specify a nonseparating
curve by a pair of genus-1 separating curves. The key to using sharing pairs
to define the extension is showing that certain sharing pairs are preserved
under incidence-preserving maps and that the image is independent of our
choice of the sharing pair.

#### Step 2 (sharing pairs are preserved).

An incidence-preserving map $\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$
sends certain sharing pairs to sharing pairs (Theorem 5.2). In order to prove
this, we also show that $\phi$ preserves the genus of each curve in $Y^{s}$
(Theorem 5.1). The proofs of these statements are intertwined: cases of
Theorem 5.2 are used to prove cases of Theorem 5.1 which are used to prove
other cases of Theorem 5.2, and so on. A key tool in this analysis is the
characterization of sharing pairs given in [BM04, Lem. 4.1].

#### Step 3 (extending from $Y^{s}$ to $Y$).

Any incidence-preserving map $\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$
extends to a locally-injective map
$\widetilde{\phi}:Y\rightarrow\mathcal{C}(\Sigma)$ (Theorem 6.1). Given
$\phi$, for $y\in Y\setminus Y^{s}$, we define $\widetilde{\phi}(y)$ to be the
curve shared by $\phi(\alpha),\phi(\beta)$, where $\alpha,\beta$ are a sharing
pair for $y$. We need to prove that $\widetilde{\phi}$ is well-defined,
simplicial, and locally injective. The proofs of these statements build on the
results of Step 2. An important ingredient is to show that $\widetilde{\phi}$
preserves (certain) chains of nonseparating curves (Lem. 6.6).

Steps 2 and 3 are fairly involved and are at the technical heart of the paper.

######  Remark 1.6 (Comparison with [BM04]).

Steps 2 and 3 above are similar to steps in an argument of Brendle–Margalit
[BM04, §4] (part of which was corrected by Kida [Kid13, §5]) that proves that
an incidence-preserving map
$\mathcal{C}^{s}(\Sigma)\rightarrow\mathcal{C}^{s}(\Sigma)$ extends to an
incidence preserving map $\mathcal{C}(\Sigma)\rightarrow\mathcal{C}(\Sigma)$.
One difference in replacing the domain $\mathcal{C}^{s}(\Sigma)$ by a finite
subcomplex $Z$ is that this greatly reduces the flexibility of certain
constructions. As a simple example if $\alpha\neq\beta$ are disjoint
separating curves, then it is easy to prove that there is another separating
curve $\gamma$ with $i(\alpha,\gamma)\neq 0$ and $i(\beta,\gamma)=0$. But this
fact is not generally true if $\alpha,\beta,\gamma$ are restricted to a finite
subcomplex $Z\subset\mathcal{C}^{s}(\Sigma)$. This is related to the fact that
an incidence-preserving map on a finite subcomplex is not generally injective
(c.f. Remark 1.1).

#### Acknowledgements.

The authors thank C. Leininger and D. Margalit for several helpful comments on
a draft of this paper.

##  2 Background and terminology

###  2.1 Curves and curve complexes

An essential simple closed curve $\alpha\subset\Sigma$ is an embedded circle
that does not bound a disk. The _curve complex_ $\mathcal{C}(\Sigma)$ is the
simplicial complex with a vertex for each isotopy class of essential, simple
closed curve on $\Sigma$ and a $k$-simplex for every collection of $(k+1)$
isotopy classes that can be realized disjointly on $\Sigma$. See e.g. [FM12,
§4.1]. Below ``curve" always means ``essential simple closed curve". We will
usually not distinguish between a curve an its isotopy class.

A curve $\alpha\subset\Sigma$ is separating if $\Sigma\setminus\alpha$ is
disconnected. The _genus_ of a separating curve $\alpha\subset\Sigma$ is the
minimum genus of the two components of $\Sigma\setminus\alpha$. The
_separating curve complex_ $\mathcal{C}^{s}(\Sigma)\subset\mathcal{C}(\Sigma)$
is defined as the subcomplex spanned by separating curves.

For a pair of isotopy classes of curves $\alpha,\beta$, the _geometric
intersection number_ $i(\alpha,\beta)$ is the minimum of $|a\cap b|$ over all
representatives $a\in\alpha$ and $b\in\beta$. See [FM12, §1.2]. Whenever
working with curves $\alpha,\beta\subset\Sigma$, we always assume that they
are in _minimal position_ , i.e. realize the geometric intersection number of
their isotopy classes. If $i(\alpha,\beta)=0$, we say that $\alpha$ and
$\beta$ are _disjoint_ ; otherwise we say that $\alpha$ and $\beta$
_intersect_.

A _chain_ is a sequence of curves $(x_{1},\ldots,x_{n})$ such that

$i(x_{j},x_{k})=\begin{cases}1&|k-j|=1\\\ 0&\text{else}\end{cases}.$

Given a chain, we use $N(x_{1}\cup\cdots\cup x_{n})$ to denote a regular
neighborhood. The boundary $\partial N(x_{1}\cup\cdots\cup x_{n})$ is a simple
closed curve when $n$ is even, and a bounding pair when $n$ is odd. We will
frequently use chains to define curves on $\Sigma$; furthermore, in our
pictures we will often indicate a curve by drawing a chain defining it, rather
than drawing the curve itself.

###  2.2 Locally injective and incidence-preserving maps

A simplicial map $f:Z_{1}\rightarrow Z_{2}$ is _locally injective_ if the
restriction to the star of every vertex is injective. When $Z_{2}$ is a curve
complex and $Z_{1}\subset Z_{2}$ is a subcomplex, this criterion in particular
implies that if $\alpha\neq\beta$ are disjoint curves, then also
$f(\alpha)\neq f(\beta)$ are disjoint curves.

Given a subcomplex $Z\subset\mathcal{C}^{s}(\Sigma)$, a simplicial map
$\phi:Z\rightarrow\mathcal{C}^{s}(\Sigma)$ is _incidence-preserving_ if
$i(\alpha,\beta)\neq 0$ implies that $i(\phi(\alpha),\phi(\beta))\neq 0$ for
$\alpha,\beta\in Z$. Note that for a simplicial map $i(\alpha,\beta)=0$
implies that $i(\phi(\alpha),\phi(\beta))=0$, so for incidence preserving maps
we have $i(\alpha,\beta)=0$ if and only if $i(\phi(\alpha),\phi(\beta))=0$.

The proof of the following lemma is easy using the definitions; see [Irm04,
Lem. 3.1]

######  Lemma 2.1.

Fix $Z\subset\mathcal{C}^{s}(\Sigma)$. Assume that whenever $\alpha,\beta\in
Z$ and $i(\alpha,\beta)=0$ then exists $\gamma\in Z$ so that
$i(\alpha,\gamma)=0$ and $i(\beta,\gamma)\neq 0$. Then any incidence-
preserving map $Z\rightarrow\mathcal{C}^{s}(\Sigma)$ is injective.

###  2.3 Sharing pairs and spines

For a nonseparating curve $y\subset\Sigma$, we say that curves
$\alpha_{1},\alpha_{2}$ are a _sharing pair that shares $y$_ if (1)
$\alpha_{1},\alpha_{2}$ are genus-1 curves; (2) if $\Sigma_{j}$ denotes the
genus-1 subsurfaces bounded by $\alpha_{j}$, then $\Sigma_{1}\cap\Sigma_{2}$
is an annulus containing $y$; and (3) the complement of
$\Sigma_{1}\cup\Sigma_{2}$ in $\Sigma$ is connected.

It is often convenient to specify a sharing pair by a choice of _spine_. A
triple $(x_{1},y,x_{2})$ of curves is called a _spine_ if $i(x_{j},y)=1$ and
$i(x_{1},x_{2})\leq 1$. If $(x_{1},y,x_{2})$ is a spine, then
$\alpha_{i}:=\partial N(x_{i}\cup y)$ defines a sharing pair. We call
$(x_{1},y,x_{2})$ a spine of the sharing pair $\alpha_{1}$ and $\alpha_{2}$.
For example, a chain of length 3 is the spine of a sharing pair.

######  Remark 2.2.

We will only ever need to use spines that are actually chains. Nevertheless,
we will continue to use this more general notion to make it easier to
reference the results of [BM04] that we use.

The following lemma from [BM04, Lem. 4.1] gives a characterization of sharing
pairs in terms of genus and intersection data of curves. The subsequent
Corollary 2.5 gives a checkable condition for showing that an incidence-
preserving map preserves sharing pairs.

######  Lemma 2.3 (Brendle–Margalit).

Assume $\Sigma$ has genus $\geq 4$. Let $\alpha,\beta$ be genus-$1$ separating
curves in $\Sigma$. Then $(\alpha,\beta)$ is a sharing pair if and only if
there exist separating curves $w,x,y,z$ in $\Sigma$ with the following
properties: $z$ bounds a genus-$2$ subsurface $\Sigma_{z}$; $\alpha$ and
$\beta$ are in $\Sigma_{z}$ and intersect each other; $x$ and $y$ are
disjoint; $w$ intersects $z$, but not $a$ and not $b$; $x$ intersects $a$ and
$z$, but not $b$; $y$ intersects $b$ and $z$, but not $a$.

$\alpha$ at 66 2535 $\beta$ at 40 2525 $w$ at 160 2540 $x$ at 200 2548 $y$ at
100 2553 $z$ at 115 2510

Figure 1: Curves $\alpha,\beta$ and $w,x,y,z$ as in the statement of Lemma
2.3.

We say that the curves $w,x,y,z$ _certify_ that $(\alpha,\beta)$ are a sharing
pair.

######  Remark 2.4.

The condition that $z$ has genus 2 is the only place where genus $g\geq 4$ is
needed in Lemma 2.3 (for $g=3$, all separating curves have genus 1).

######  Corollary 2.5.

Let $Z\subset\mathcal{C}^{s}(\Sigma)$ be a subcomplex, and fix an incidence-
preserving map $\phi:Z\rightarrow\mathcal{C}^{s}(\Sigma)$. Let
$\alpha,\beta\in Z$ be a sharing pair certified by $w,x,y,z\in Z$. Suppose
that $\phi(\alpha),\phi(\beta)$ are genus-$1$ curves, that $\phi(z)$ is a
genus-$2$ curve, and that $\phi(\alpha)$ and $\phi(\beta)$ are in the
genus-$2$ subsurface bounded by $\phi(z)$. Then $\phi(\alpha),\phi(\beta)$ is
a sharing pair.

###### Proof.

By Lemma 2.3, it suffices to show that $\phi(x),\phi(y),\phi(z),\phi(w)$
certify that $\phi(\alpha),\phi(\beta)$ is a sharing pair. This is true
because $\phi(\alpha)$ and $\phi(\beta)$ are in the genus-2 subsurface bounded
by $\phi(z)$ by assumption and because incidence-preserving maps preserve
whether two curves are disjoint or intersect. ∎

###  2.4 (Extended) mapping class group

The _mapping class group_
$\operatorname{Mod}(\Sigma):=\pi_{0}(\text{Homeo}^{+}(\Sigma))$ is the group
of isotopy classes of orientation-preserving homeomorphisms of $\Sigma$. The
_extended mapping class group_
$\operatorname{Mod}^{\pm}(\Sigma)=\pi_{0}(\text{Homeo}(\Sigma))$ is an index-2
super-group that includes orientation-reversing mapping classes. It is
necessary for us to consider $\operatorname{Mod}^{\pm}(\Sigma)$ because the
automorphism group of $\mathcal{C}(\Sigma)$ is
$\operatorname{Mod}^{\pm}(\Sigma)$ [Iva97].

##  3 The finite rigid set of separating curves

In this section we give the construction of the subcomplex
$Y^{s}\subset\mathcal{C}^{s}(\Sigma)$ in Theorem A. The complex $Y^{s}$ will
be defined as $Y\cap\mathcal{C}^{s}(\Sigma)$ for a certain subcomplex
$Y\subset\mathcal{C}(\Sigma)$.

We define $Y$ as the subcomplex spanned by a certain set of vertices. For ease
of exposition, we will not distinguish between $Y$ and its set of vertices. In
particular, we will define

$Y=C\cup S\cup B\cup U\cup V,$

where $C,S,B,U,V$ are certain collections of curves on $\Sigma$. We proceed
now to define these curves.

#### Chain, separating, and bounding pair curves.

Let $C=\\{c_{0},\ldots,c_{2g+1}\\}$ denote the chain of curves in Figure 2.

$c_{0}$ at 260 2595 $\cdots$ at 410 2595 $c_{1}$ at 315 2575 $c_{2}$ at 340
2605 $c_{3}$ at 365 2575 $c_{2g-1}$ at 440 2575 $c_{2g}$ at 500 2595
$c_{2g+1}$ at 400 2615 $s_{\\{0,1\\}}$ at 620 2575 $b_{[0,4]}^{+}$ at 690 2610
$b_{[0,4]}^{-}$ at 690 2570

Figure 2: Left: chain curves in $C$. Right: sample curves in $S$ and $B$.

It is convenient to identify
$\\{0,\ldots,2g+1\\}\cong\mathbb{Z}/(2g+2)\mathbb{Z}$, and define an
_interval_ $J\subset\\{0,\ldots,2g+1\\}$ to be a sequence of consecutive
equivalence classes. Sometimes we write $[i,j]$ for the interval
$\\{i,i+1,\ldots,j\\}$. For an interval $J$, define $c_{J}=\bigcup_{j\in
J}c_{j}$ and

$s_{J}=\partial N(c_{J}).$

When $|J|\geq 2$ is even, $s_{J}$ is a separating curve of genus $|J|/2$. When
$|J|>1$ is odd, $s_{J}$ is a bounding pair. Note that the union of the even
chain curves $c_{0}\cup c_{2}\cup\dots\cup c_{2g}$ separates $\Sigma$ into
$\Sigma_{e}^{+}$ and $\Sigma_{e}^{-}$. Similarly, the union of the odd chain
curves $c_{1}\cup c_{3}\cup\dots\cup c_{2g+1}$ cuts $\Sigma$ into
$\Sigma_{o}^{+}$ and $\Sigma_{o}^{-}$. Let $b_{J}^{+}$ be the curve in $s_{J}$
that lies in $\Sigma_{o}^{+}$ or $\Sigma_{e}^{+}$, and let $b_{J}^{-}$ be the
curve in $s_{J}$ that lies in $\Sigma_{o}^{-}$ or $\Sigma_{e}^{-}$. Then
$s_{J}=b_{J}^{+}\cup b_{J}^{-}$. Write $S=\\{s_{J}:|J|\text{ even}\\}$ and
$B=\\{b_{J}^{\pm}:|J|\text{ odd}\\}$ for the collection of these curves. See
Figure 2 for an example.

#### Genus-1 curves.

Next we consider the following collection $U$ of genus-1 curves. Fix $J=[i,j]$
with $|J|$ is odd. Noting that
$i(b_{J}^{\pm},c_{i-1})=1=i(b_{J}^{\pm},c_{j+1})$, we define

$u_{i-1,J}^{\pm}=\partial N(c_{i-1}\cup b_{J}^{\pm})\>\>\>\text{ and
}\>\>\>u_{j+1,J}^{\pm}=\partial N(c_{j+1}\cup b_{J}^{\pm}),$

and we let $U$ denote the collection of all such curves. Figure 3 contains an
example.

$u_{3,[4,6]}^{+}$ at 320 2675 $v_{[3,4]*[4,8]*\\{9\\}}^{+}$ at 690 2670

Figure 3: Sample curves in $U$ (left) and $V$ (right).

#### Genus-2 curves.

Finally, we consider the following collection $V$ of genus-2 curves. These
curves are obtained by taking $v=\partial N(x_{1}\cup\cdots\cup x_{4})$, where
$x_{1},\ldots,x_{4}$ is a chain of length 4 with three of the $x_{j}$ in $C$
and one in $B$. There are various cases depending on the position of the curve
in $B$ in the chain.

To make this precise, write $J=[i,j]$, where $|J|$ is odd. A _predecessor
interval_ (resp.  _successor interval_) for $J$ is an interval of the form
$[*,i-1]$ or $[i-1,*]$ (resp. $[j+1,*]$ or $[*,j+1]$). If $\pi$ and $\sigma$
are predecessor and successor intervals, respectively, and $|\pi|+|\sigma|=3$,
then we we define

$v_{\pi*J*\sigma}^{\pm}=\partial N(c_{\pi}\cup b_{J}^{\pm}\cup c_{\sigma}),$
(1)

and we define $V$ as the collection of all such curves. Note that we allow one
of $\pi$ or $\sigma$ to be empty. The assumption $|\pi|+|\sigma|=3$ implies
that $v_{\pi*J*\sigma}^{\pm}$ is a genus-2 curve. See Figure 3 for an example.

Now we define

$Y=C\cup S\cup B\cup U\cup V$ (2)

and

$Y^{s}=S\cup U\cup V$ (3)

######  Remark 3.1.

It is easy to check that $Y^{s}$ satisfies the assumption of Lemma 2.1, so we
observe that any incidence-preserving map
$\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$ is injective.

The following is a more precise statement of Theorem A.

######  Theorem 3.2.

Fix $g\geq 4$. Let $Y^{s}\subset\mathcal{C}^{s}(\Sigma_{g})$ be the subcomplex
defined by (3). Any incidence-preserving map
$\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$ is induced by a mapping class.

#### Symmetries of $Y$.

To prove Theorem 3.2, it will be helpful to observe that $Y$ has several
symmetries. This will significantly reduce the casework in our arguments.

######  Lemma 3.3.

There are mapping classes $r,s\in\operatorname{Mod}(\Sigma)$ that preserve
$C=\\{c_{0},\ldots,c_{2g-1}\\}$ and act on $C$ by

$r(c_{i})=c_{i+1}\>\>\>\>\text{ and }s(c_{i})=c_{-i-2}$

with indices interpreted modulo $2g+2$.

Here $s$ acts on $C$ as the ``reflection" fixing $c_{g}$ and $c_{2g+1}$.

###### Proof of Lemma 3.3.

The homeomorphism $s$ is pictured in Figure 4; the description depends on
whether the genus of $\Sigma$ is odd or even.

$s$ at 670 2700 $\cdots$ at 625 2760 $\cdots$ at 715 2760 $\cdots$ at 848 2760
$\cdots$ at 972 2760 $s$ at 910 2700 $\bar{r}$ at 1095 2700

Figure 4: Symmetries of $\Sigma$ and $\Sigma/\langle\iota\rangle$

To define $r$, observe that $C$ is invariant by a hyperelliptic involution
$\iota$. The image of $C$ on $\Sigma/\langle\iota\rangle\cong S^{2}$ is a
sequence of arcs between the branch points, and an order-($2g+2$) rotation
$\bar{r}$ of $S^{2}$ that permutes the branched points, as pictured in Figure
4, lifts to a homeomorphism $r$ of $\Sigma$ with the desired properties. ∎

######  Remark 3.4.

By the construction of $Y$, a mapping class that preserves $C$ will preserve
$Y$. Therefore, we can also view $r,s$ as automorphisms of $Y$. They generate
a dihedral group. In addition, there is an obvious hyperelliptic involution
that preserves $Y$. It acts trivially on $C$, but interchanges
$b^{+}\leftrightarrow b^{-}$ and $u^{+}\leftrightarrow u^{-}$ for $b^{\pm}\in
B$ and $u^{\pm}\in U$.

##  4 $Y$ is rigid in $\mathcal{C}(\Sigma)$

In this section we start the proof of Theorem A, as outlined in the
introduction for the set $Y^{s}$ defined in §3. In what follows $\Sigma$
denotes a fixed closed surface of genus $g\geq 4$.

######  Proposition 4.1.

Let $Y\subset\mathcal{C}(\Sigma)$ be the subcomplex defined in (2). Any
locally injective map $\Phi:Y\rightarrow\mathcal{C}(\Sigma)$ is induced by an
extended mapping class.

Using the notation of §3, we set

$X=C\cup S\cup B\cup U.$ (4)

This is the finite rigid set of [AL13, Thm. 5.1].

######  Theorem 4.2 (Aramayona–Leininger).

Let $X\subset\mathcal{C}(\Sigma)$ be the subcomplex defined in (4). Any
locally injective map $\Phi:X\rightarrow\mathcal{C}(\Sigma)$ is induced by an
extended mapping class.

###### Proof of Proposition 4.1.

Recall from [AL13, Defn. 2.1] that a vertex $v$ of $\mathcal{C}(\Sigma)$ is
_uniquely determined_ by a $A\subset\mathcal{C}(S)$ if $v$ is the unique
isotopy class of curve that can be realized disjointly from every curve in
$A$.

Claim. Each $v\in V$ is uniquely determined by a subset $A\subset X$.

First we explain why the claim implies the proposition. Let
$\Phi:Y\rightarrow\mathcal{C}(\Sigma)$ be a locally injective map. Since
$Y\supset X$ and $X$ is rigid by [AL13, Thm. 5.1] (see Theorem 4.2 above),
there exists $h\in\operatorname{Mod}^{\pm}(\Sigma)$ such that
$h\bigr{|}_{X}=\Phi\bigr{|}_{X}$. To show $h\bigr{|}_{Y}=\Phi$, it remains to
show that $h(v)=\Phi(v)$ for $v\in V$. By the claim, for $v\in V$, there
exists $A\subset X$ such that $v$ is uniquely determined by $A$. Since $\Phi$
is simplicial, $\Phi(v)$ is disjoint from $\Phi(A)$. Since
$h\in\operatorname{Mod}^{\pm}(\Sigma)$, there is a unique vertex
$v^{\prime}\in\mathcal{C}(\Sigma)$ that is disjoint from $h(A)$, and
$h(v)=v^{\prime}$. Thus $\Phi(v)=v^{\prime}=h(v)$, as desired.

Proof of Claim. Fix $v=v_{\pi*J*\sigma}^{\epsilon}$ in $V$, where
$\epsilon\in\\{\pm\\}$. Using the automorphisms defined in Lemma 3.3, we can
assume that $|\pi|<|\sigma|$ and $J=[2,\ldots,2k]$. For concreteness we take
$\epsilon=+$; the case $\epsilon=-$ is similar.

We want to find a collection of curves in $X$ that fill
$\Sigma_{0}:=S\setminus v$. In fact, we will show that there are two chains in
$\Sigma_{0}$ of length $4$ and $2(g-2)$ that fill the two components of
$\Sigma_{0}$ (which have genus $2$ and $g-2$). For the chain of length-4, we
use the chain used to define $v$. A choice for the other chain is pictured in
Figure 5.

$...$ at 112 2405 $...$ at 267 2405 $...$ at 490 2405 $...$ at 645 2406 $...$
at 109 2317 $...$ at 265 2317 $...$ at 486 2318 $...$ at 642 2318

Figure 5: A chain of length $2g-4$ that fills the complement of the chain of
length 4. Top row: the case $|\pi|=1$. Bottom row: the case $|\pi|=0$.

∎

##  5 Incidence-preserving maps $Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$

The goal of this section is to prove the following two theorems, which say
that incidence-preserving maps $\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$
preserve genus and (certain) sharing pairs.

######  Theorem 5.1 (Genus is preserved).

Fix $Y^{s}$ as defined in (3), and let
$\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$ be an incidence-preserving map.
For $\alpha\in Y^{s}$, the curves $\alpha$ and $\phi(\alpha)$ have the same
genus.

######  Theorem 5.2 (Sharing pair is preserved).

Let $\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$ be an incidence-preserving
map, and let $(\alpha,\beta)$ be a sharing pair with one of the following
spines.

  1. (i)

$(c_{i},c_{i+1},c_{i+2})$

For any interval $[i,j]$ of odd length $\geq 3$,

  1. (ii)

$(c_{i-2},c_{i-1},b_{[i,j]}^{\pm})$

  2. (iii)

$(c_{i},c_{i-1},b_{[i,j]}^{\pm})$

  3. (iv)

$(b_{[i,j]}^{\pm},c_{j+1},c_{j+2})$

  4. (v)

$(b_{[i,j]}^{\pm},c_{j+1},c_{j})$

  5. (vi)

$(c_{i-1},b_{[i,j]}^{\pm},c_{j+1})$

Then $(\phi(\alpha),\phi(\beta))$ is also a sharing pair.

We prove these results in a ``bootstrapping" fashion. We prove, in order, the
following statements.

  1. (A)

The genus is preserved for curves in $X^{s}$.

  2. (B)

Sharing pairs of type (i) are preserved.

  3. (C)

The genus is preserved for $v_{\pi*J*\sigma}^{\pm}\in V$ such that either
$\pi$ or $\sigma$ is empty.

  4. (D)

Sharing pairs of type (ii)–(v) are preserved.

  5. (E)

The genus is preserved for $v_{\pi*J*\sigma}^{\pm}\in V$ such that both $\pi$
and $\sigma$ are nonempty.

  6. (F)

Sharing pairs of type (vi) are preserved.

Each statement is used to prove the next statement. Together these statements
prove Theorems 5.1 and 5.2.

We begin with the proofs. Fix an incidence-preserving map
$\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$.

Proof of (A) Fix $\alpha\in X^{s}$. We want to show that $\alpha$ and
$\phi(\alpha)$ have the same genus. Recall that $X^{s}=S\cup U$. Up to a
permutation of $X$ (c.f. Remark 3.4), we can assume $\alpha$ belongs to the
collection $\Delta$ of curves in Figure 6, which form a maximal simplex of
$X^{s}$, and also a maximal simplex of $\mathcal{C}^{s}(\Sigma)$. The curve
$\alpha$ gives a decomposition
$\Delta=\Delta_{L}\cup\\{\alpha\\}\cup\Delta_{R}$ into curves on either side
of $\alpha$. (If $\alpha$ has genus 1, then one of $\Delta_{L}$ or
$\Delta_{R}$ is empty.)

$\cdots$ at 490 2360

Figure 6: Curves in $\Delta$

Claim. The curves in $\phi(\Delta_{L})$ lie on the same side of
$\phi(\alpha)$, and similarly for $\phi(\Delta_{R})$.

Given the claim, we deduce that $\alpha$ and $\phi(\alpha)$ have the same
genus as follows. The genus of a separating curve $\gamma$ is determined by
the maximum number of disjoint separating curves that fit on either of the two
components of $\Sigma\setminus\gamma$. The claim implies that this number is
the same for $\alpha$ and $\phi(\alpha)$ unless $\phi(\Delta_{L})$ and
$\phi(\Delta_{R})$ lie on the same side of $\phi(\alpha)$. This is not an
issue when $\alpha$ has genus 1 (since then one of $\Delta_{L}$ or
$\Delta_{R}$ is empty). Assume that $\alpha$ has genus $\geq 2$ and suppose
for a contradiction that $\phi(\Delta_{L})$ and $\phi(\Delta_{R})$ lie on the
same side of $\phi(\alpha)$. Since $\phi(\Delta_{L})$ is disjoint from
$\phi(\Delta_{R})$, this implies that $\phi(\alpha)$ has genus 1. This implies
that $\phi(\Delta)$ has more genus-1 curves than $\Delta$, but this is
impossible because every maximal collection of separating curves on $\Sigma$
has exactly $g$ curves with genus $1$.

To prove the claim, it suffices to observe that for each pair
$x,x^{\prime}\in\Delta_{L}$, there exists $y\in Y^{s}$ that intersects both
$x$ and $x^{\prime}$ and such that $y$ is disjoint from $\alpha$. This is easy
to check and implies that $\phi(y)$ intersects both $\phi(x)$ and
$\phi(x^{\prime})$, and $\phi(y)$ is disjoint from $\phi(\alpha)$. Thus
$\phi(x)$ and $\phi(x^{\prime})$ lie on the same side of $\phi(\alpha)$. The
same argument works for $\Delta_{R}$.

Proof of (B) The strategy for proving cases of Theorem 5.2 is the same in
every case. By Corollary 2.5, it suffices to find curves $w,x,y,z\in Y^{s}$
that certify that $(\alpha,\beta)$ is a sharing pair and show the following
properties: $\phi(\alpha)$ and $\phi(\beta)$ are genus-1 curves; $\phi(z)$ is
a genus-2 curve; $\phi(\alpha)$ and $\phi(\beta)$ lie in the genus-2
subsurface bounded by $\phi(z)$. It is easy enough to find $w,x,y,z$. But to
know the genus of $\phi(\alpha),\phi(\beta),\phi(z)$, we need $\alpha,\beta,z$
to belong to the list of cases for which Theorem 5.1 has been verified.

Fix a sharing pair $(\alpha,\beta)$ with spine $(c_{i},c_{i+1},c_{i+2})$. By a
permutation, it suffices to consider $(c_{0},c_{1},c_{2})$. Here we take
$x=c_{\\{2g,2g+1\\}}$, $y=c_{\\{3,4\\}}$, $z=c_{[0,3]}$, and
$w=c_{\\{4,5\\}}$, as in Figure 1. Note that here $\alpha=s_{\\{0,1\\}}$,
$\beta=s_{\\{1,2\\}}$, and $z=c_{[0,3]}$ all belong to $X^{s}$, so we know
that $\phi(\alpha),\phi(\beta),\phi(z)$ have the right genus by (A). Moreover,
we know that $\phi(\alpha)$ is in the genus-2 subsurface bounded by $\phi(z)$
by the proof of (A). The same is true for $\phi(\beta)$ because $\phi$ is an
incidence-preserving map and $\alpha$ intersects $\beta$. Hence by Corollary
2.5, the curves $\phi(x),\phi(y),\phi(z),\phi(w)$ certify that
$(\phi(\alpha),\phi(\beta))$ is a sharing pair.

Proof of (C) The strategy for (C) and (E) are similar. We are given a genus-2
curve $\alpha\in V$ and we want to show $\phi(\alpha)$ has genus 2. We will
reduce to showing the following two statements.

  1. ($\dagger$)

There exists a sharing pair $(\beta_{1},\beta_{2})$ that is contained in the
genus-2 component of $\Sigma\setminus\alpha$ and that belongs to the list of
sharing pairs for which Theorem 5.2 has been verified.

  2. ($\ddagger$)

There exist distinct, disjoint genus-1 curves $x_{1},\ldots,x_{g-2}$ in
$X^{s}$ that lie in the genus-$(g-2)$ component of $\Sigma\setminus\alpha$ and
such that for every pair $x_{j},x_{k}$, there exists a third curve $\gamma\in
Y^{s}$ that intersects both $x_{j}$ and $x_{k}$ and is disjoint from $\alpha$.

Once we prove $(\dagger)$ and $(\ddagger)$, we conclude that $\phi(\alpha)$
has genus 2 as follows. First, the curves $\phi(x_{1}),\ldots,\phi(x_{g-2})$
are disjoint (because $x_{1},\ldots,x_{g-2}$ are disjoint) genus-1 curves
(because $x_{1},\ldots,x_{g-2}$ are in $X^{s}$), and they lie in one component
of $\Sigma\setminus\phi(\alpha)$ (by an argument similar to the argument for
(A)). This shows that there is a component of $\Sigma\setminus\phi(\alpha)$
that has genus $\geq g-2$. Then $\phi(\alpha)$ either has genus 1 or 2.

Suppose for a contradiction that $\phi(\alpha)$ has genus 1. Consider the
sharing pair $(\beta_{1},\beta_{2})$ given in $(\ddagger)$. Since Theorem 5.2
has been verified for $(\beta_{1},\beta_{2})$, we know that
$(\phi(\beta_{1}),\phi(\beta_{2}))$ is a sharing pair. Let $\Sigma_{j}$ be the
genus-1 subsurface bounded by $\phi(\beta_{j})$, $j=1,2$. Since we assume that
$\phi(\alpha)$ has genus 1, the curves $\phi(\beta_{1}),\phi(\beta_{2})$ lie
in the component of $\Sigma\setminus\phi(\alpha)$ with genus $g-1$. Since
$\phi(\beta_{1})$ and $\phi(\beta_{2})$ are disjoint from the genus-1 curves
$\phi(x_{1}),\ldots,\phi(x_{g-2})$, we deduce that
$\phi(\beta_{1}),\phi(\beta_{2})$ lie in a subsurface $\Sigma^{\prime}$ with
genus 1 and $g-1$ boundary components. See Figure 7. One component
$\Sigma^{\prime\prime}$ of $\Sigma^{\prime}\setminus\phi(\beta_{1})$ has genus
0 and $g$ boundary components. Any arc on $\Sigma^{\prime\prime}$ with
endpoints on a single component of $\partial\Sigma^{\prime\prime}$ (such as
$\phi(\beta_{2})\cap\Sigma^{\prime\prime}$) separates $\Sigma^{\prime\prime}$,
and since the components of $\partial\Sigma^{\prime\prime}$ are separating in
$\Sigma$, it follows that $\Sigma\setminus(\Sigma_{1}\cup\Sigma_{2})$ is
disconnected, contradicting the fact that $\phi(\beta_{1}),\phi(\beta_{2})$ is
a sharing pair. Therefore, $\phi(\alpha)$ does not have genus 1.

$\phi(\alpha)$ at 1215 1890 $\phi(\beta_{1})$ at 1270 1918 $\phi(x_{1})$ at
1330 1900 $\phi(x_{2})$ at 1328 1870 $\phi(x_{g-2})$ at 1260 1860
$\Sigma^{\prime\prime}$ at 1275 1890 $...$ at 1320 1855

Figure 7:

Now we prove $(\dagger)$ and $(\ddagger)$ for $\alpha=v_{\pi*J*\sigma}^{\pm}$
with $\pi$ or $\sigma$ empty. After a permutation, we can assume
$\sigma=\varnothing$, $\pi=\\{1,2,3\\}$ and either $J=[2,2k]$ or $J=[4,2k]$.
See Figure 8.

Figure 8:

In these cases, the genus-2 component of $\Sigma\setminus\alpha$ contains the
sharing pair with spine $(c_{1},c_{2},c_{3})$, which is covered by (B).
Therefore $(\dagger)$ holds. Figure 8 also shows a choice of $g-2$ disjoint
genus-1 curves $x_{1},\ldots,x_{g-2}\in X^{s}$ in the genus-$(g-2)$ component
of $\Sigma\setminus\alpha$. Now $(\ddagger)$ is easily verified.

Proof of (D) We proceed just as in (B). For case (ii) it suffices to treat
$(c_{2},c_{3},b_{[4,2k]}^{+})$. Here we choose $x=c_{\\{0,1\\}}$,
$y=u_{2k+1,[0,2k]}^{-}$, $z=v_{\\{1,2,3\\}*[4,2k]}^{+}$, and
$w=c_{\\{2g+1,0\\}}$. See Figure 9 (top right). Observe that the curves
$\alpha,\beta\in X^{s}$ and $z\in V$ are of the form covered by (C). We also
know by the proof of (C) that $\phi(\alpha),\phi(\beta)$ are in the genus-2
subsurface bounded by $\phi(z)$ because the sharing pair
$(\phi(\alpha)=\phi(s_{\\{2,3\\}}),\phi(s_{\\{1,2\\}}))$ is in the genus-2
subsurface bounded by $\phi(z)$. Case (iii) is similar: it suffices to treat
$(c_{4},c_{3},b_{[4,2k]}^{+})$, and we choose $w,x,y,z$ in Figure 9 (bottom
right).

$w$ at 770 2130 $x$ at 730 2170 $y$ at 920 2150 $z$ at 820 2150 $w$ at 865
2035 $x$ at 840 2035 $y$ at 930 2033 $z$ at 790 2093 $...$ at 490 2168 $...$
at 605 2168

Figure 9: Top row: sharing pair of Theorem 5.2(ii). Bottom row: sharing pair
of Theorem 5.2(iii).

Finally, observe that cases (iv), (v) are equivalent to (ii) and (iii), up to
automorphisms of $X$.

Statement (E) As in (C), we proceed by proving the statements $(\dagger)$ and
$(\ddagger)$.

Fix $\alpha=v_{\pi*J*\sigma}^{\pm}$ with both $\pi$ and $\sigma$ nonempty.
After a permutation, it suffices to consider the case $\pi=\\{1\\}$,
$J=[2,2k]$, and either $\sigma=\\{2k+1,2k+2\\}$ or $\sigma=\\{2k,2k+1\\}$. See
Figure 10.

$...$ at 810 1960 $...$ at 962 1960

Figure 10:

In these cases, the genus-2 component of $\Sigma\setminus\alpha$ contains the
sharing pair with spine either $(b_{J},c_{2k+1},c_{2k+2})$ or
$(b_{J},c_{2k+1},c_{2k})$, both of which are covered by (D). Therefore
$(\dagger)$ holds. Figure 10 also shows a choice of $g-2$ disjoint genus-1
curves $x_{1},\ldots,x_{g-2}\in X^{s}$ in the genus-$(g-2)$ component of
$\Sigma\setminus\alpha$. Now $(\ddagger)$ is easily verified.

Statement (F) We proceed just as in (B) and (D). For case (vi) it suffices to
treat $(c_{1},b_{[2,2k]}^{+},c_{2k+1})$. Here we choose $x=c_{\\{0,1\\}}$,
$y=s_{\\{2k+2,2k+3\\}}$, $z=v_{\\{0,1\\}*[2,2k]*\\{2k+1\\}}^{+}$, and
$w=c_{\\{2g,2g+1\\}}$. See Figure 11. Observe that the curves $\alpha,\beta\in
X^{s}$ and $z\in V$ are of the form covered by (E). Using arguments similar to
the proof of (D) we can deduce that the curves $\phi(\alpha)$ and
$\phi(\beta)$ are in the genus-2 subsurface bounded by $\phi(z)$.

$w$ at 1766 2170 $x$ at 1490 2130 $y$ at 1695 2170 $z$ at 1510 2170 $...$ at
1180 2170 $...$ at 1320 2170 $...$ at 1562 2170

Figure 11:

##  6 Extending the incidence-preserving map

Recall that we have a defined subcomplexes $Y^{s}\subset
Y\subset\mathcal{C}(\Sigma)$ in (2) and (3). In this section we prove the last
step in the proof of Theorem A:

######  Theorem 6.1.

Any incidence-preserving map $\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$
admits a locally injective extension
$\widetilde{\phi}:Y\rightarrow\mathcal{C}(\Sigma)$.

###  6.1 Defining the extension $\widetilde{\phi}$

Fix an incidence-preserving map
$\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$. Since $Y=Y^{s}\cup C\cup B$,
we want to define $\widetilde{\phi}(y)$ for $y\in C\cup B$.

For $y\in B$, there is a unique sharing pair $\alpha,\beta$ with spine
$(x,y,z)$ with $x,z\in C$. By Theorem 5.2, $\phi(\alpha),\phi(\beta)$ is a
sharing pair, so we define $\widetilde{\phi}(y)$ to be the curve shared by
$\phi(\alpha),\phi(\beta)$.

For $y\in C$, choose a sharing pair $\alpha,\beta$ with spine $(x,y,z)$ such
that $x,z\in C\cup B$ and at most one of $x,z$ is in $B$. Again
$\phi(\alpha),\phi(\beta)$ is also a sharing pair by Theorem 5.2, and we
define $\widetilde{\phi}(y)$ to be the curve shared by
$\phi(\alpha),\phi(\beta)$.

When $y\in C$, there are multiple choices for the spine $(x,y,z)$ and hence
for $\alpha,\beta$. As such, we need to show that $\widetilde{\phi}$ is well-
defined, independent of this choice. For this, we follow the approach from
[BM04, §4.2]. Two spines $(x,y,z)$ and $(x,y,z^{\prime})$ are said to _differ
by a move_ if $(z,y,z^{\prime})$ is also a spine. The proof of the following
lemma is exactly the same as [BM04, Lem. 4.4].

######  Lemma 6.2.

Let $\phi:Y^{s}\rightarrow\mathcal{C}^{s}(\Sigma)$ be an incidence-preserving
map, and fix sharing pairs $(\alpha,\beta)$ and $(\alpha,\beta^{\prime})$ with
spines $(x,y,z)$ and $(x,y,z^{\prime})$ that differ by a move. Assume that
$\alpha,\beta,\beta^{\prime},x,y,z,z^{\prime}$ are curves in $Y$. Assume that
(i) there exists $\gamma\in Y^{s}$ such that $\gamma$ intersects $z^{\prime}$
but intersects none of $x,y,z$, and (ii) each of $(\phi(\alpha),\phi(\beta))$,
$(\phi(\alpha),\phi(\beta^{\prime}))$, and
$(\phi(\beta),\phi(\beta^{\prime}))$ are sharing pairs. Then
$(\phi(\alpha),\phi(\beta))$ and $(\phi(\alpha),\phi(\beta^{\prime}))$ share
the same curve.

Now we can show $\widetilde{\phi}$ is well defined on $C$. First note that any
two spines used in the definition of $\widetilde{\phi}(y)$ for $y\in C$ differ
by at most two moves (they all differ by a single move from the (unique) spine
consisting only of chain curves). It is easy to check that condition (i) in
Lemma 6.2 holds; see e.g. Figure 12.

$x$ at 700 2530 $y$ at 740 2507 $z$ at 765 2518 $z^{\prime}$ at 870 2555
$\gamma$ at 850 2500 $\cdots$ at 810 2530 $\cdots$ at 955 2530

Figure 12: Proving that $\widetilde{\phi}$ is well-defined.

Finally, condition (ii) in Lemma 6.2 holds by Theorem 5.2. Therefore,
$\widetilde{\phi}$ is well defined.

######  Remark 6.3.

Having defined $\widetilde{\phi}$ on the vertices of $Y$, to show that
$\widetilde{\phi}$ extends simplicially to the subcomplex spanned by these
vertices (which we are also denoting by $Y$), it suffices to show that if
$\alpha,\beta\in Y$ are disjoint, then
$\widetilde{\phi}(\alpha),\widetilde{\phi}(\beta)$ are also disjoint. This is
proved in Proposition 6.4 below.

###  6.2 The extension $\widetilde{\phi}$ is locally-injective

This section contains the final step in the proof of Theorem A. We need to
show that $\widetilde{\phi}$ is simplicial (c.f. Remark 6.3) and that
$\widetilde{\phi}$ is injective on the star of each vertex $v\in Y$. These
statements follow from Propositions 6.4 and 6.5 below (which in fact show that
$\widetilde{\phi}$ is injective on $Y$).

######  Proposition 6.4.

If $\alpha\neq\beta$ are vertices in $Y$ and $i(\alpha,\beta)=0$, then
$\widetilde{\phi}(\alpha)\neq\widetilde{\phi}(\beta)$ and
$i(\widetilde{\phi}(\alpha),\widetilde{\phi}(\beta))=0$.

######  Proposition 6.5.

If $\alpha,\beta$ are vertices in $Y$ and $i(\alpha,\beta)\neq 0$, then
$\widetilde{\phi}(\alpha)\neq\widetilde{\phi}(\beta)$.

The proofs of Propositions 6.4 and 6.5 rely on the following lemma.

######  Lemma 6.6 (Chains under $\widetilde{\phi}$).

Fix $x_{1},x_{2}\in X\setminus X^{s}\equiv C\cup B$.

  1. (i)

Assume that $x_{1}$ and $x_{2}$ do not form a bounding pair. If $x_{1}$ and
$x_{2}$ are distinct and disjoint, then $\widetilde{\phi}(x_{1})$ and
$\widetilde{\phi}(x_{2})$ are distinct and disjoint.

  2. (ii)

Assume that at most one of the $x_{1},x_{2}$ belongs to $B$. Then if
$i(x_{1},x_{2})=1$, then
$i(\widetilde{\phi}(x_{1}),\widetilde{\phi}(x_{2}))=1$.

###### Proof.

$None$ It suffices to observe that there exists $y_{1},y_{2}\in C$ so that
$i(x_{j},y_{k})=\delta_{jk}$ (Kronecker delta). For then one can find
$z_{1},z_{2}\in C$ so that $(y_{j},x_{j},z_{j})$ is the spine for a sharing
pair $(\alpha_{j},\beta_{j})$ for $x_{j}$ (this is illustrated in Figure 13);
by construction $\alpha_{1}=\partial N(x_{1}\cup y_{1})$ and
$\alpha_{2}=\partial N(x_{2}\cup y_{2})$ are distinct and disjoint. Then so
too are $\widetilde{\phi}(\alpha_{1})$ and $\widetilde{\phi}(\alpha_{2})$ (by
Remark 3.1), and from this we deduce that $\widetilde{\phi}(x_{1})$ and
$\widetilde{\phi}(x_{2})$ are disjoint and distinct because they lie in the
disjoint genus-1 subsurfaces bounded by $\widetilde{\phi}(\alpha_{1})$ and
$\widetilde{\phi}(\alpha_{2})$ (by Theorem 5.2).

$x_{1}$ at 1078 2535 $y_{1}$ at 1130 2555 $z_{1}=z_{2}$ at 1155 2507 $y_{2}$
at 1180 2555 $x_{2}$ at 1205 2520 $\uparrow$ at 1155 2522

Figure 13:

$None$ There are two cases to consider: either $x_{1},x_{2}\in C$ or $x_{1}\in
C$ and $x_{2}\in B$.

First assume that $x_{1},x_{2}\in C$. Up to a permutation of $X$, we can
assume that $x_{1}=c_{1}$ and $x_{2}=c_{2}$. Consider the chain
$(x_{0},x_{1},x_{2},x_{3},x_{4})$ in Figure 14 (left).

$x_{0}$ at 1078 2440 $x_{1}$ at 1130 2458 $x_{2}$ at 1155 2425 $x_{3}$ at 1180
2458 $x_{4}$ at 1205 2425 $x_{0}$ at 1340 2440 $x_{1}$ at 1388 2417 $x_{2}$ at
1460 2453 $x_{3}$ at 1518 2420 $x_{4}$ at 1555 2427 $\cdots$ at 1458 2435

Figure 14:

Each triple $(x_{j-1},x_{j},x_{j+1})$ is a spine of a sharing pair
$\alpha_{j},\beta_{j}$ for $x_{j}$ (note with this notation that
$\alpha_{j+1}=\beta_{j}$). By Theorem 5.2 and the definition of
$\widetilde{\phi}$, $\phi(\alpha_{j}),\phi(\beta_{j})$ is a sharing pair for
$\widetilde{\phi}(x_{j})$. Denoting $\Sigma_{j}$ the genus-1 component of
$\Sigma\setminus\widetilde{\phi}(\beta_{j})$ for $j=1,2$, we deduce that
$\widetilde{\phi}(x_{j})$ and $\widetilde{\phi}(x_{j+1})$ are contained in
$\Sigma_{j}$ (for example, $\widetilde{\phi}(\beta_{1}=\alpha_{2})$ forms a
sharing pair with both $\widetilde{\phi}(\alpha_{1})$ and
$\widetilde{\phi}(\beta_{2})$, which implies that $\Sigma_{1}$ contains both
$\widetilde{\phi}(x_{1})$ and $\widetilde{\phi}(x_{2})$).

Since $\widetilde{\phi}(x_{1})$ and $\widetilde{\phi}(x_{3})$ are distinct and
disjoint by part (i) of the lemma, it follows that
$\widetilde{\phi}(x_{1})\neq\widetilde{\phi}(x_{2})$ and
$\widetilde{\phi}(x_{3})\neq\widetilde{\phi}(x_{2})$. This implies that
$\widetilde{\phi}(x_{1}),\widetilde{\phi}(x_{3})$ both intersect
$\widetilde{\phi}(x_{2})$ (distinct curves on a torus with one boundary
component must intersect).

Since $\phi(\alpha_{2}=\beta_{1}),\phi(\beta_{2})$ is a sharing pair for
$\widetilde{\phi}(x_{2})$, the subsurface $\Sigma_{1}\cup\Sigma_{2}$ is
obtained from an annular neighborhood $A$ of $\widetilde{\phi}(x_{2})$ by
attaching two rectangles as pictured in Figure 15.

$\widetilde{\phi}(x_{1})$ at 818 2290 $\widetilde{\phi}(x_{2})$ at 855 2325
$\widetilde{\phi}(x_{3})$ at 930 2288 $A$ at 745 2284

Figure 15: The subsurface $\Sigma_{1}\cup\Sigma_{2}$ is homeomorphic to an
annulus $A$ with two rectangles attached along the boundary.

Observe that if $i(\widetilde{\phi}(x_{1}),\widetilde{\phi}(x_{2}))>1$, then
$\widetilde{\phi}(x_{1})$ and $\widetilde{\phi}(x_{3})$ are not disjoint,
which is a contradiction (this is because the condition
$i(\widetilde{\phi}(x_{1}),\widetilde{\phi}(x_{2}))>1$ implies that
$\widetilde{\phi}(x_{1})\cap A$ separates the two intervals in $\partial A$
where the rectangle for $\Sigma_{2}$ is attached). From this we conclude that
$i(\widetilde{\phi}(x_{1}),\widetilde{\phi}(x_{2}))=1$, as desired.

The proof in the case $x\in C$ and $y\in B$ is entirely similar, except we
work with the chain $(x_{0},\ldots,x_{4})$ pictured in Figure 14 (right). ∎

###### Proof of Proposition 6.4.

We consider cases depending on whether or not $\alpha,\beta$ are separating,
i.e. whether or not $\alpha,\beta\in Y^{s}$.

#### Case 1:

$\alpha\in Y^{s}$ and $\beta\in Y^{s}$. We already know that
$\widetilde{\phi}(\alpha)$ and $\widetilde{\phi}(\beta)$ are disjoint because
$\widetilde{\phi}\bigr{|}_{Y^{s}}=\phi$, and $\phi$ is incidence preserving.
Furthermore, $\widetilde{\phi}(\alpha)\neq\widetilde{\phi}(\beta)$ by Remark
3.1.

#### Case 2:

$\alpha\in Y^{s}$ and $\beta\notin Y^{s}$.

First observe that it suffices to find a sharing pair $\gamma,\delta$ for
$\alpha$ such that $(\gamma,\delta)$ is on the list in Theorem 5.2 and
$\gamma$ is disjoint from $\beta$. For then $\phi(\gamma),\phi(\delta)$ is a
sharing pair for $\widetilde{\phi}(\alpha)$ (by Theorem 5.2 and by definition
of $\widetilde{\phi}$), which implies in particular that $\phi(\gamma)$ is a
genus-1 curve with $\widetilde{\phi}(\alpha)$ contained in the genus-1
component of $\Sigma\setminus\phi(\gamma)$, and since
$\widetilde{\phi}(\beta)=\phi(\beta)$ is a separating curve disjoint from
$\phi(\gamma)$, this implies that $\phi(\beta)$ lies in the genus-$(g-1)$
component of $\Sigma\setminus\phi(\gamma)$. Therefore
$\widetilde{\phi}(\alpha)$ and $\widetilde{\phi}(\beta)$ are both disjoint and
distinct since they lie in different components of
$\Sigma\setminus\phi(\gamma)$.

#### Case 2(i):

Assume that $\alpha\in C$. Without loss of generality $\alpha=c_{1}$. Consider
$(c_{0},c_{1},c_{2})$, which is the spine of a sharing pair for
$\alpha=c_{1}$. If $\beta$ is disjoint from either $c_{0}$ or $c_{1}$, then we
are done.

Claim. If $\beta$ intersects both $c_{0}$ and $c_{2}$, then $\beta$ has one of
the following forms (see also Figure 16)

$\partial N(c_{3}\cup c_{4}\cup b_{[0,2]}^{\pm}\cup c_{2g+1})\>\>\>\text{ or
}\>\>\>\partial N(c_{3}\cup b_{[0,2]}^{\pm}\cup c_{2g+1}\cup
c_{2g})\>\>\>\text{ or }\>\>\>\partial N(c_{1}\cup b_{[2,2k]}^{\pm})$ $\text{
or }\>\>\>\partial N(c_{1}\cup b_{[2,2k]}^{\pm}\cup c_{2k+1}\cup
c_{2k})\>\>\>\text{ or }\>\>\>\partial N(c_{1}\cup b_{[2,2k]}^{\pm}\cup
c_{2k+1}\cup c_{2k+2}).$

The details of this claim are not hard to check: since $\beta\in S\cup U\cup
V$, we can write $\beta=\partial N(x_{1}\cup\cdots\cup x_{\ell})$, where
$\ell$ is even, the $x_{j}$ belong to $C\cup B$, and at most one $x_{j}$ is in
$B$; in order for $\beta$ to intersect $c_{0}$ and $c_{2}$, at least one of
the $x_{j}$ has to intersect $c_{0}$ and similarly for $c_{2}$, and none of
the $x_{j}$ can be equal to $c_{0}$ or $c_{2}$; examining the possibilities
yields the claim.

$c_{0}$ at 1395 2520 $c_{2}$ at 1470 2530 $\beta$ at 1505 2500 $...$ at 1605
2520 $...$ at 1675 2430

Figure 16: Curves $\beta$ in $Y^{s}$ that intersect both $c_{0}$ and $c_{2}$.
In the bottom figure, one can either ignore the two green curves or include
two adjacent green curves in defining $\beta$.

Observe that in every case there is a different spine $(c_{0},c_{1},\star)$
for a sharing pair for $\alpha=c_{1}$ with $\beta$ disjoint from $\star$, so
again we are done. In the first two examples in Figure 16, one can choose
$\star=b_{[2,4]}^{-}$. In the last case, one can take $\star=b_{[3,j]}^{+}$.

#### Case 2(ii):

Assume that $\alpha\in B$. Without loss of generality $\alpha=b_{[2,2k]}^{+}$.
Consider the spine $(c_{1},\alpha,c_{2k+1})$ for a sharing pair for $\alpha$
(note: there is only one sharing pair for $\alpha$ in $Y^{s}$). As in Case
2(i), if $\beta$ disjoint from either $c_{1}$ or $c_{2k+1}$, then we are done,
so assume $\beta$ intersects both $c_{1}$ and $c_{2k+1}$. Examining the cases,
we conclude that either

$\beta=\partial N(c_{0}\cup b_{[1,2k+1]}^{\pm}\cup c_{2k+2}\cup c_{2k+3})$

or

$\beta=\partial N(c_{2g+1}\cup c_{0}\cup b_{[1,2k+1]}^{\pm}\cup c_{2k+2}).$

See the Figure 17.

$c_{1}$ at 1570 2200 $c_{2k+1}$ at 1665 2200 $\alpha$ at 1620 2230 $\beta$ at
1715 2190 $...$ at 1622 2210 $...$ at 1942 2210

Figure 17: Curves $\beta$ in $Y^{s}$ that intersect both $c_{1}$ and
$c_{2k+1}$ and are disjoint from $\alpha$.

These cases can be treated similarly. For concreteness, take the former
definition of $\beta$. By Lemma 6.6 and Theorem 5.1, the image of the chain
$(c_{0},b_{[1,2k+1]}^{+},c_{2k+2},c_{2k+3})$ under $\widetilde{\phi}$ is also
a chain and is supported in the genus-2 component of
$\Sigma\setminus\phi(\beta)$. Since $\widetilde{\phi}(\alpha)$ is disjoint
from the curves in this chain (again by Lemma 6.6), we deduce that $\alpha$ is
supported in the genus-$(g-2)$ component of $\Sigma\setminus\phi(\beta)$. This
shows $\widetilde{\phi}(\alpha)$ and $\widetilde{\phi}(\beta)$ are disjoint.
(They are necessarily distinct because one is separating and the other is
nonseparating.)

#### Case 3:

$\alpha\notin Y^{s}$ and $\beta\notin Y^{s}$.

The case when $\alpha$ and $\beta$ do not form a bounding pair is treated in
Lemma 6.6. Assume then that $\alpha$ and $\beta$ form a bounding pair. Without
loss of generality, assume $\alpha=b_{[2,2k]}^{+}$ and $\beta=b_{[2,2k]}^{-}$.

First we show that $i(\widetilde{\phi}(\alpha),\widetilde{\phi}(\beta))=0$.
Consider the chain $c=(c_{0},\ldots,c_{2g})$. By Lemma 6.6,
$\widetilde{\phi}(c)$ is also a chain and

$i(\widetilde{\phi}(\alpha),\widetilde{\phi}(c_{j}))=i(\widetilde{\phi}(\beta),\widetilde{\phi}(c_{j}))=\begin{cases}1&j=1,2k+1\\\
0&\text{else}\end{cases}$

Since $\widetilde{\phi}(c)$ is a chain of length $2g+1$, its complement
$\Sigma\setminus\widetilde{\phi}(c)$ is a union of two disks; similarly

$\Sigma\setminus\big{[}\widetilde{\phi}(c_{0})\cup\widetilde{\phi}(c_{2})\cup\cdots\cup\widetilde{\phi}(c_{2g})\big{]}$
(5)

is a union of two genus-0 surfaces, each with $g+1$ boundary components. Since
$\widetilde{\phi}(\alpha)$ and $\widetilde{\phi}(\beta)$ are disjoint from
each $\widetilde{\phi}(c_{2k})$, they define curves on (5). If
$\widetilde{\phi}(\alpha)$ and $\widetilde{\phi}(\beta)$ lie in different
components of (5), we are done. If $\widetilde{\phi}(\alpha)$ and
$\widetilde{\phi}(\beta)$ lie on the same component, then they must be
isotopic (there is only one curve on the punctured sphere with the given
intersection data with the $\widetilde{\phi}(c_{j})$). In any case, we have
shown that $i(\widetilde{\phi}(\alpha),\widetilde{\phi}(\beta))=0$.

It remains to show that $\widetilde{\phi}(\alpha)$ and
$\widetilde{\phi}(\beta)$ are actually distinct. Suppose for a contradiction
that $\widetilde{\phi}(\alpha)=\widetilde{\phi}(\beta)$. Consider the curves

$\gamma=\partial
N(\underbrace{b_{[2,2k]}^{+}}_{\alpha},c_{2k+1},c_{2k+2},c_{2k+3})\>\>\>\text{
and }\>\>\>\delta=\partial N(\underbrace{b_{[2,2k]}^{-}}_{\beta},c_{2k+1}).$

See Figure 18.

$\alpha$ at 1700 2020 $\beta$ at 1700 1980 $\gamma$ at 2020 2018 $\delta$ at
2020 1984 $...$ at 1703 2000 $...$ at 2020 2000

Figure 18: Left: chain curves $c_{0},\ldots,c_{2g}$ and the curves
$\alpha,\beta$. Right: chains used to define $\gamma$ and $\delta$.

On the one hand, $\gamma,\delta\in Y^{s}$ intersect, so
$\widetilde{\phi}(\gamma)=\phi(\gamma)$ and
$\widetilde{\phi}(\delta)=\phi(\delta)$ also intersect. By Theorem 5.1 and
Lemma 6.6, $\phi(\delta)$ has genus 1 and
$\widetilde{\phi}(\beta),\widetilde{\phi}(c_{2k+1})$ are a chain in the
genus-1 component of $\Sigma\setminus\phi(\delta)$. Similarly, $\phi(\gamma)$
is a genus-2 curve that contains the chain
$(\widetilde{\phi}(\alpha),\widetilde{\phi}(c_{2k+1}),\widetilde{\phi}(c_{2k+2}),\widetilde{\phi}(c_{2k+3}))$
in the genus-2 component of $\Sigma\setminus\phi(\gamma)$. In particular,
$\widetilde{\phi}(\beta)=\widetilde{\phi}(\alpha)$ and
$\widetilde{\phi}(c_{2k+1})$ are disjoint from $\widetilde{\phi}(\gamma)$, but
this implies that $\phi(\delta)=\partial
N(\widetilde{\phi}(\beta)\cup\widetilde{\phi}(c_{2k+1}))$ is disjoint from
$\phi(\beta)$, a contradiction. This shows that
$\widetilde{\phi}(\alpha)\neq\widetilde{\phi}(\beta)$, as desired. ∎

###### Proof of Proposition 6.5.

If $\alpha$ and $\beta$ are both separating, then the proposition follows from
the fact that $\widetilde{\phi}$ extends $\phi$, and $\phi$ is incidence-
preserving. If exactly one of $\alpha,\beta$ is separating, then the
proposition holds because then exactly one of
$\widetilde{\phi}(\alpha),\widetilde{\phi}(\beta)$ is separating.

Now consider the case $\alpha,\beta\in C\cup B$ and both nonseparating. If
$\alpha$, say, belongs to $C$, then we can apply Lemma 6.6 to conclude that
$i(\widetilde{\phi}(\alpha),\widetilde{\phi}(\beta))=1$, which implies that
$\widetilde{\phi}(\alpha)\neq\widetilde{\phi}(\beta)$. If both $\alpha$ and
$\beta$ belong to $B$, then $\alpha$ and $\beta$ do not form a bounding pair
because we assume that $\alpha$ and $\beta$ intersect. This implies that
$\alpha$ and $\beta$ have a different intersection pattern with the curves in
$C$. Since this intersection pattern is preserved under $\widetilde{\phi}$ by
Lemma 6.6, this implies $\widetilde{\phi}(\alpha)\neq\widetilde{\phi}(\beta)$.
∎

##  7 A finite rigid set of separating curves for $g=3$

In this section, let $\Sigma$ be a closed oriented surface of genus 3. As is
observed in §1, we can take a single vertex in $\mathcal{C}^{s}(\Sigma)$ to be
a finite rigid set because every separating curve in $\Sigma$ is genus-1.
However, larger finite rigid sets may also exist and identifying them may shed
some light on Question 1.4 and Question 1.5. We will present explicitly
another finite rigid set $X^{s}\subset\mathcal{C}^{s}(\Sigma)$ similar to the
ones constructed in §3 for $g\geq 4$. The homotopy type of $X^{s}$ is
determined as well.

The collections of curves $C,S,B$ and $U$ on $\Sigma$ are defined as in §3.
Let

$X=C\cup S\cup B\cup U.$ (6)

and

$X^{s}=S\cup U$ (7)

In fact $X$ is the finite rigid set for $\Sigma$ defined in [AL13] and
$X^{s}=X\cap\mathcal{C}^{s}(\Sigma)$. Here is the main results of this
section.

######  Theorem 7.1.

Let $\Sigma$ be a closed, oriented surface of genus $3$ and set
$X^{s}\subset\mathcal{C}^{s}(\Sigma)$ be the subcomplex defined by (7). Then
the simplicial complex $X^{s}$ has the homotopy type of a wedge of $21$
circles, and any incidence-preserving map
$\phi:X^{s}\to\mathcal{C}^{s}(\Sigma)$ is induced by a mapping class.

###### Proof.

The same method used above to prove Theorem 3.2 can be applied to verify that
$X^{s}$ is rigid. Below we prove that $X^{s}$ is homotopy equivalent to a
wedge of 21 circles.

First we understand the vertices of $X^{s}$, starting with $S\subset X^{s}$.
(For this, it is helpful to recall the notation from §3.) Note that every
curve $s_{J}\in S$ has the form $s_{\\{i,i+1\\}}$, and there are eight of
these curves. To see this, first observe that in genus 3, there are $8$ chain
curves, so curves in $S$ have the form $s_{J}$ where $|J|\in\\{2,4,6\\}$. If
$|J|=6$, then $s_{J}$ is null-homotopic, and if $|J|=4$, then
$s_{J}=s_{J^{\prime}}$ where $|J^{\prime}|=2$ (concretely, if $J=[j,j+3]$,
then $J^{\prime}=\\{j+5,j+6\\}$). For simplicity, we denote

$s_{i}:=s_{\\{i,i+1\\}}\>\>\>\text{ for }\>\>\>i=0,\ldots,7.$

Similarly, we find that the curves in $U\subset X^{s}$ all have the form

$u_{i}^{\pm}:=\partial N(c_{i}\cup b_{\\{i+1,i+2,i+3\\}}^{\pm})\>\>\>\text{
for }\>\>\>i=0,\ldots,7.$

(Note in particular that
$b_{\\{i+1,i+2,i+3\\}}^{\pm}=b_{\\{i-3,i-2,i-1\\}}^{\pm}$ for each $i$.)
Therefore $X^{s}=S\cup U$ has 24 vertices.

Next we consider the edges in $X^{s}$. It is easy to check the following
incidence relations

$i(s_{k},s_{j})\text{ is }\begin{cases}=0&\text{if }|k-j|>2\\\ \neq
0&\text{else}\end{cases}\hskip 36.135pti(u_{k},s_{j})\text{ is
}\begin{cases}=0&\text{if }j=k+2\text{ or }j=k-3\\\ \neq
0&\text{else}\end{cases}$

and

$i(u_{k}^{\pm},u_{j}^{\epsilon})\text{ is }\begin{cases}=0&\text{if }j=k\pm
2\text{ and }\epsilon=\mp\\\ \neq 0&\text{else}\end{cases}$

Here $|k-j|$ denotes the distance in
$\mathbb{Z}/8\mathbb{Z}=\\{0,\ldots,7\\}$. For checking this, it is helpful to
observe that if $a=\partial N(x\cup y)$ and $b=\partial N(z\cup w)$ are
distinct genus-1 curves (here $x,y$ and $z,w$ are chains of length 2), then
$i(a,b)=0$ if and only if $x\cup y$ is disjoint from $z\cup w$.

From the intersection data, we find that the subcomplex $S\subset X^{s}$ is
1-dimensional (see Figure 19 (left)), and that the subcomplex $U\subset X^{s}$
is the disjoint union of 4 simplicial circles, each with 4 vertices. These
circles are pictured in Figure 20.

Finally, each vertex of $U$ is connected to two vertices of $S$, and together
these three vertices span a 2-simplex in $X^{s}$. See Figure 19 (right).
Observe that the edges $\\{s_{i},s_{i+5}\\}$ each belong to two 2-simplices
(the other vertices are $u_{i+3}^{\pm}$), while each edge connecting a vertex
in $U$ to a vertex in $S$ belongs to a single 2-simplex. Therefore, each
2-simplex of $X^{s}$ has a ``free edge", which implies that $X^{s}$
deformation retracts to a 1-dimensional complex. The simplicial complex
$X^{s}$ has 24 vertices, 60 edges and 16 faces, so it has Euler characteristic

$\chi(X^{s})=24-60+16=-20$

and we conclude that $X^{s}$ is homotopy equivalent to a wedge of 21 circles.

$s_{0}$ at 180 1700 $s_{1}$ at 260 1700 $s_{5}$ at 180 1490 $s_{4}$ at 260
1490 $s_{7}$ at 105 1630 $s_{6}$ at 105 1560 $s_{2}$ at 335 1630 $s_{3}$ at
335 1560 $u_{7}^{+}$ at 582 1620 $u_{3}^{+}$ at 582 1564 $u_{5}^{-}$ at 555
1595 $u_{1}^{-}$ at 610 1595 $s_{1}$ at 540 1695 $s_{4}$ at 620 1695 $s_{5}$
at 540 1490 $s_{0}$ at 620 1490 $s_{7}$ at 475 1630 $s_{2}$ at 475 1560
$s_{3}$ at 690 1630 $s_{6}$ at 690 1560

Figure 19: Left: the subcomplex $S\subset X^{s}$. Right: the circle
$\\{a,b,c,d\\}$ is connected to $S$ by four 2-simplices.

∎

Figure 20: Each frame is a circle in $U$ with four vertices. In total, $U$ is
the disjoint union of these circles.

## References

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Junzhi Huang
Peking University
<EMAIL_ADDRESS>
Bena Tshishiku
Department of Mathematics, Brown University
<EMAIL_ADDRESS>


amsmathForeign command inputencinputenc package ignored with utf8 gensymbNot
defining inkscapelatex=false,inkscapearea=page

††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Strong Thermomechanical Noise Squeezing Stabilized by Feedback

Aida Mashaal  Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17,
2100, Copenhagen, Denmark  Center for Hybrid Quantum Networks, Niels Bohr
Institute, University of Copenhagen, Blegdamsvej 17, 2100, Copenhagen, Denmark
Lucio Stefan  Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17,
2100, Copenhagen, Denmark  Center for Hybrid Quantum Networks, Niels Bohr
Institute, University of Copenhagen, Blegdamsvej 17, 2100, Copenhagen, Denmark
Andrea Ranfagni  Niels Bohr Institute, University of Copenhagen, Blegdamsvej
17, 2100, Copenhagen, Denmark  Center for Hybrid Quantum Networks, Niels Bohr
Institute, University of Copenhagen, Blegdamsvej 17, 2100, Copenhagen, Denmark
Letizia Catalini  Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich,
Switzerland  Quantum Center, ETH Zürich, 8093 Zürich, Switzerland  Ilia
Chernobrovkin  Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17,
2100, Copenhagen, Denmark  Center for Hybrid Quantum Networks, Niels Bohr
Institute, University of Copenhagen, Blegdamsvej 17, 2100, Copenhagen, Denmark
Thibault Capelle  Niels Bohr Institute, University of Copenhagen, Blegdamsvej
17, 2100, Copenhagen, Denmark  Center for Hybrid Quantum Networks, Niels Bohr
Institute, University of Copenhagen, Blegdamsvej 17, 2100, Copenhagen, Denmark
Eric C. Langman  Niels Bohr Institute, University of Copenhagen, Blegdamsvej
17, 2100, Copenhagen, Denmark  Center for Hybrid Quantum Networks, Niels Bohr
Institute, University of Copenhagen, Blegdamsvej 17, 2100, Copenhagen, Denmark
Albert Schliesser<EMAIL_ADDRESS>Niels Bohr Institute,
University of Copenhagen, Blegdamsvej 17, 2100, Copenhagen, Denmark  Center
for Hybrid Quantum Networks, Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, 2100, Copenhagen, Denmark

###### Abstract

Squeezing the quadrature noise of a harmonic oscillator enhances its
sensitivity. Stabilization of the anti-squeezed quadrature can overcome the 3
dB limit of squeezing by parametric modulation. Here, we apply this method to
soft-clamped membrane resonators, which hold promise for both classical and
quantum sensing applications. We compare piezo and capacitive parametric
modulation, and observe thermomechanical squeezing by up to 17 dB and 21 dB,
respectively. Finally, we provide a full quantum theory of a combination of
this approach with quantum-limited motion measurement and conclude that
quantum squeezing is feasible at moderate cryogenic temperatures and realistic
device parameters.

##  I Introduction

Mechanical resonators can sense a variety of physical entities—including force
[Eichler2022], acceleration [Gerberding2015], mass [Chaste2012], voltage
[Bagci2014] and magnetic fields [Simonsen2019]—with astounding sensitivity. As
technical improvements continue to advance their performance, sensors now
approach limits dictated by quantum mechanics. For example, most common
techniques to read out a mechanical resonator position add noise that is
subject to the standard quantum limit (SQL) [Thorne1992, Aspelmeyer2014].
However, techniques to overcome the latter have been demonstrated recently
[Mason2019, Yu2020].

In addition, the sensitivity can be limited by fluctuations intrinsic to the
mechanical resonator, primarily its thermal fluctuations and then ultimately
its vacuum fluctuations (zero-point motion). When such limits are reached, a
route to improve the sensitivity consists of squeezing the measured mechanical
quadrature noise. This is possible through parametric modulation of the
elastic constant, as demonstrated early on with cantilever force sensors
[Rugar1991]. Recently, this principle has been extended to the quantum regime
for single trapped ions [Burd2019]. The latter constitutes an impressive
conceptual achievement, even if concrete sensing applications are less
obvious. Quantum squeezing has also been achieved with superconducting
micromechanical resonators cooled to dilution refrigerator temperatures, using
microwave reservoir engineering [wollman2015, pirk2015, lecocq2015].

In contrast, our goal is to demonstrate strong parametric squeezing—ultimately
squeezing also quantum noise—in a versatile sensor platform, namely soft-
clamped nanomechanical membrane resonators. They offer large sensing areas on
the order of $(0{.}3\leavevmode\nobreak\ \mathrm{mm})^{2}$, combined with
nanogram effective masses and quality factors that can exceed $10^{9}$
[Seis2022]. Since their recent advent [Tsaturyan_2017], they have already been
used for nanoscale force imaging [Halg2021], measurement of quantum
fluctuations of optical forces [Rossi_2018, Huang2024], and are promising
candidates for sensing minute forces e.g. from individual spins [Eichler2022,
Kolkowitz2012, Kosata2020].

Given the usually large thermal motion undergone by such resonators even at
cryogenic temperatures, a potential limitation of conventional squeezing
protocols is that in the steady state, parametric systems reach an oscillation
instability when the target quadrature is squeezed by $3\,\mathrm{dB}$.
However, this limit can be overcome using e.g. a detuned parametric drive
[Szorkovszky2013], or by actively stabilizing the anti-squeezed quadrature
[Vinante2013, Pontin2014]. In a proof-of-principle experiment, the latter
protocol has been used to demonstrate squeezing of up to $15\,\mathrm{dB}$
[Poot2015].

In this work, we benchmark a similar protocol applied to our membrane force
sensor, and compare the performance of two different parametric modulation
methods: one based on piezo actuation and the other on capacitive coupling. By
applying quadrature feedback stabilization, we demonstrate a maximum squeezing
of $17\,\mathrm{dB}$ and of $21\,\mathrm{dB}$, respectively, with moderate
device parameters at room temperature. Furthermore, we develop a model that
predicts the behavior of the protocol in the quantum regime under continuous
quantum-limited monitoring. We estimate that, with realistic parameters and at
a moderate bath temperature $T=10\,\mathrm{K}$, squeezing below the zero-point
fluctuations is attainable.

Figure 1: Experimental protocol and platform for capacitive parametric
driving. (a) A sketch of the experimental protocol implemented to create
classical squeezed states of the mechanical motion. A mechanical resonator in
a thermal state (i) is displaced along the amplitude quadrature $X_{1}$ by a
resonant drive while the phase quadrature $X_{2}$ is stabilized through a
phase-locked loop (PLL), as depicted in (ii). The $X_{2}$ quadrature is
therefore dampened by a feedback rate $\Gamma_{\mathrm{fb}}\propto
g_{\mathrm{fb}}$ (purple arrows). By applying a parametric drive, the mean
value of the $X_{1}$ quadrature is deamplified, while the variance is squeezed
by a rate $\Gamma_{\mathrm{s}}\propto g_{\mathrm{s}}$ (orange arrows), shown
in (iii). The anti-squeezing of the $X_{2}$ quadrature variance is
counteracted by the PLL, preventing the system from undergoing self-
oscillation. (b) A double-defect silicon nitride (SiN) membrane lies on
aluminum (Al) pillars of a silicon (Si) chip with a printed gold (Au)
electrode circuit. An aluminum pad is deposited onto one of the defects,
forming a capacitor with the underlying electrodes. For detection purposes, a
hole is etched on the side of the electrode chip beneath the non-metallized
defect. (c) A rendered angled view of the capacitive platform with an inset of
a microscopic image.

##  II Protocol

A mechanical oscillator, characterized by an effective mass $m$, a mechanical
damping rate $\Gamma_{\mathrm{m}}$ and a natural elastic constant
$k_{\mathrm{m}}$ is driven around resonance by a force
$F_{\mathrm{d}}=F_{0}\cos\left(\Omega t\right)$. The angular frequency
$\Omega$ of the drive is set by an electronic oscillator which is phase-locked
to the mechanical motion. The parametric drive, offset by a phase
$\phi_{\mathrm{p}}$ relative to the electronic oscillator, modulates the
mechanical stiffness at twice the drive frequency $2\Omega$ with an amplitude
$k_{\mathrm{p}}$.

We express the apparent mechanical motion, referenced to the phase-locked
electronic oscillator, as

$\ddot{x}+\Gamma_{\mathrm{m}}\dot{x}+\frac{1}{m}\left[k_{\mathrm{m}}+k_{\mathrm{p}}\sin{\left(2\Omega
t+\phi_{\mathrm{p}}\right)}\right]x=\frac{F_{\mathrm{th}}+F_{\mathrm{d}}+F_{\mathrm{fb}}}{m}\;,$
(1)

where $F_{\mathrm{th}}$ is the thermomechanical noise force and
$F_{\mathrm{fb}}$ is an effective force representing the action of the phase-
locked loop that stabilizes the relative phase between the mechanical
oscillation and the electronic oscillator drive.

For a high quality factor oscillator
($Q=\Omega_{\mathrm{m}}/\Gamma_{\mathrm{m}}\gg 1$) and
$\Omega\sim\Omega_{\mathrm{m}}$, the motion is described in the frame rotating
at the frequency $\Omega$ in terms of the two slow-varying quadratures
$X_{1}(t)$ and $X_{2}(t)$, defined as:

$x(t)=X_{1}(t)\sin\left(\Omega t\right)+X_{2}(t)\cos\left(\Omega t\right)\;.$
(2)

When the parametric modulation and the stabilization are turned off, the
oscillator is driven exclusively by the thermomechanical noise and the drive
$F_{\mathrm{d}}$. The phase quadrature $X_{2}$ as a function of the frequency
difference $\left(\Omega-\Omega_{\mathrm{m}}\right)$ shows a dispersive shape,
thus providing an error signal suitable for the frequency stabilization.

With both the feedback and the parametric modulation turned on, the phase
space vector—whose components are represented by a sum of a coherent amplitude
$\bar{X_{i}}$ and a fluctuating component $\delta X_{i}$—fluctuates around the
equilibrium position
$\left(\bar{X}_{1},\bar{X}_{2}\right)^{T}=\left(F_{0}/m\,\Omega_{\mathrm{m}}\Gamma_{\mathrm{m}}\left(1+g_{\mathrm{s}}\right),0\right)^{T}$,
where $g_{\mathrm{s}}=\Gamma_{s}/\Gamma_{\mathrm{m}}$ and
$\Gamma_{\mathrm{s}}=k_{\mathrm{p}}/(2m\Omega_{\mathrm{m}})$ is the squeezing
rate. The equations for the residual fluctuations of the two quadratures in
the Fourier domain read [briant2003]:

$\displaystyle\delta\tilde{X_{1}}\left(\omega\right)=\frac{1}{2m\Omega_{\mathrm{m}}}\,\frac{1}{-i\omega+\frac{\Gamma_{\mathrm{m}}}{2}\left(1+g_{\mathrm{s}}\right)}\tilde{F}_{\mathrm{th}2}\left(\omega\right)\;,$
(3)
$\displaystyle\delta\tilde{X_{2}}\left(\omega\right)=-\frac{1}{2m\Omega_{\mathrm{m}}}\,\frac{1}{-i\omega+\frac{\Gamma_{\mathrm{m}}}{2}\left(1-g_{\mathrm{s}}+g_{\mathrm{fb}}\right)}\tilde{F}_{\mathrm{th}1}\left(\omega\right)\;,$

where $g_{\mathrm{fb}}=\Gamma_{\mathrm{fb}}/\Gamma_{\mathrm{m}}$, with the
feedback force expressed as
$F_{\mathrm{fb}}=m\Omega_{\mathrm{m}}\Gamma_{\mathrm{fb}}X_{2}\sin{\left(\Omega
t\right)}$, and the thermal force
$F_{\mathrm{th}}\left(t\right)=F_{\mathrm{th1}}\left(t\right)\sin\left(\Omega
t\right)+F_{\mathrm{th2}}\left(t\right)\cos\left(\Omega t\right)$. From Eq.
(3) it follows that the two quadratures have different mechanical
susceptibilities. The amplitude quadrature $X_{1}$, which is in-phase with the
parametric modulation, experiences an increased damping rate. Contrarily, as
the squeezing increases, the damping of the phase quadrature $X_{2}$ is
reduced. The system reaches the instability threshold when
$\Gamma_{\mathrm{s}}=\Gamma_{\mathrm{m}}$, thereby limiting the maximum
squeezing of $\delta\bar{X}_{1}$ to $3\,\mathrm{dB}$, unless the feedback is
turned on.

Figure 2: Parametric modulation of a membrane resonator using a piezo
actuator. (a) Amplitude gain of the $\bar{X}_{1}$ quadrature as a function of
the applied parametric drive, for $\phi_{p}=\pi/2$ (red dots) and $\phi_{p}=0$
(blue dots). When the quadrature is amplified it diverges at the threshold
(dashed gray line), while the $3\,\mathrm{dB}$ limit can be overcome (gray
shaded area) when $X_{1}$ is deamplified. The inset shows the tuning of the
resonance frequency as a function of DC piezo voltage. All data are down-
sampled for visual clarity. The solid line are fits to the experimental data.
(b) Power spectral density of the $X_{1}$ quadrature for increasing parametric
drive voltage. When the parametric drive voltage is increased from
$0\,\mathrm{V}$ (yellow line) to large values (increasingly dark lines), the
total mode damping increases. (c) Variance $\sigma_{1}^{\;2}$ obtained from
fitting the PSDs in panel (b) normalized by the variance at
$V_{\mathrm{p}}=0\,\mathrm{V}$, $\sigma_{0}^{\;2}$. The markers have the same
color coding as in panel (b). The solid line is a fit to the experimental
data.

The thermal force quadratures are uncorrelated and the double-sided power
spectral density (PSD) reads
$S_{F_{\mathrm{th1}}}=S_{F_{\mathrm{th2}}}=2S_{F_{\mathrm{th}}}$, with
$S_{F_{\mathrm{th}}}=2m\Gamma_{\mathrm{m}}k_{\mathrm{B}}T$. For the system at
equilibrium with the thermal bath, the variances for the two quadratures are:

$\left(\frac{\sigma_{1}}{\sigma_{0}}\right)^{2}=\frac{1}{1+g_{\mathrm{s}}}\;,\qquad\left(\frac{\sigma_{2}}{\sigma_{0}}\right)^{2}=\frac{1}{1-g_{\mathrm{s}}+g_{\mathrm{fb}}}\;,$
(4)

where $\sigma_{0}^{2}=k_{\mathrm{B}}T/m\,\Omega_{\mathrm{m}}^{2}$.

As such, the protocol allows to get a displaced squeezed state from an initial
thermal state. Thanks to the tracking of the mechanical resonance, it is
possible to squeeze the amplitude fluctuations below the standard 3 dB limit.

##  III Experimental Results

In this work, we employ highly-stressed $\mathrm{Si_{3}N_{4}}$ soft-clamped
membrane resonators patterned with a phononic crystal. The resonators are
operated at room temperature and, in order to minimize the effect of gas
damping, at a pressure of $10^{-7}\,\mathrm{mbar}$. We devise two alternative
configurations to drive the membranes parametrically. In the first
configuration, we use a membrane hosting a single defect in the center. The
membrane is glued on the four sides of the silicon frame to a ring
piezoelectric actuator (Piezomechanik, HPCh 150/15-8/3). When a voltage is
applied to the actuator, its vertical expansion (contraction) is accompanied
by a contraction (expansion) in the horizontal plane. As a result, the
membrane experiences a modulation of the stress, and therefore of the
resonance frequency, via the shear force applied to the frame, resulting in
parametric driving.

The second assembly we explore is illustrated in Fig. 1(b). The membrane hosts
a phononic dimer, which gives rise to a symmetric and an anti-symmetric mode
whose envelopes span both defects [Catalini2020]. One of the two defects is
metallized with a $50\,\mathrm{nm}$-thick aluminum pad, with a diameter of
$100\,\mathrm{\upmu m}$ [Seis2022]. The membrane is then bonded to a silicon
chip with a printed circuit consisting of a circular signal electrode
surrounded by a grounding rim. The metallized defect forms a capacitor with
the underlying electrodes. A voltage $V(t)$ applied to the circuit creates a
modulation of the elastic constant:

$k_{\mathrm{p}}=-\frac{1}{2}\left.\left(\frac{\partial^{2}C(x)}{\partial
x^{2}}\right)\right|_{x_{\mathrm{eq}}}V^{2}(t)\;,$ (5)

where $C(x)$ is the total capacitance and $x_{\mathrm{eq}}$ is the equilibrium
position of the membrane.

The mechanical motion is detected using a Mach-Zender interferometer operating
at a wavelength of $1535\,\mathrm{nm}$ [Catalini2020]. The interference
signal, measured by a balanced photodetector, is demodulated by a lock-in
amplifier (HF2LI, Zurich Instruments) at the mechanical resonance frequency.
The lock-in amplifier also provides the signals to drive the mechanical
resonance and to apply the parametric modulation.

In order to follow the protocol described in the previous section, the
membrane is first driven at resonance. The demodulated signal has a phase
offset from the resonant force given by the overall delay caused by the
electronics. The resonant force phase is then optimized so that the
demodulated quadratures correspond to the phase ($X_{2}$) and amplitude
($X_{1}$) quadratures. A PLL within the lock-in amplifier provides the
feedback needed to stabilize the phase quadrature by adjusting the driving
frequency. The latter is frequency-doubled and the resulting signal is sent to
the actuator to apply the parametric drive.

### Parametric drive with a piezo

We first study the response of the resonator to the drive scheme shown in Fig.
1(a). The resonator design uses a so called Dahlia-class defect [Rossi_2018]
centered in a phononic crystal structure approximately $3.3\,\mathrm{mm}\times
3.3\,\mathrm{mm}$ in size, etched into a $41\,\mathrm{nm}$ membrane of
stoichiometric LPCVD $\mathrm{Si_{3}N_{4}}$ with a stress of 1.25 GPa. The
resonator has a frequency $\Omega_{\mathrm{m}}=2\pi\times 1.33\,\mathrm{MHz}$
and a quality factor $Q=\left(28\pm 5\right)\times 10^{6}$. To certify the
quality of the assembly, we first measure the tuning of the resonance
frequency with the DC voltage applied to the ring piezo actuator. The shift is
linear, with a slope that is typically $\simeq-300\,\mathrm{Hz/V}$ (inset of
Fig. 2(a)).

After turning on the resonant drive and the PLL for the stabilization scheme,
we systematically increase the strength of the parametric modulation
$V_{\mathrm{p}}$ (i.e. the peak voltage of the signal applied to the piezo
actuator) while recording the $X_{1}$ quadrature. The phase of the parametric
drive is chosen to get either deamplification ($\phi_{\mathrm{p}}=0$) or
amplification ($\phi_{\mathrm{p}}=\pm\pi/2$) of the amplitude of
$\bar{X}_{1}$. The resulting amplitude gain curves, with the gain defined as
$|\bar{X}_{1}/\bar{X}_{1,V_{\mathrm{p}}=0}|$, are plotted in Fig. 2(a). When
$\bar{X}_{1}$ is amplified, the feedback (acting only on $X_{2}$) does not
provide stabilization and the quadrature can reach the parametric threshold,
which occurs when $g_{\mathrm{s}}\equiv V_{\mathrm{p}}/V_{\mathrm{th}}=1$,
where $V_{\mathrm{th}}$ is the threshold voltage. On the contrary, when the
parametric phase is set such that $X_{1}$ is the quadrature undergoing
squeezing, the threshold can be overcome and a maximum quadrature amplitude
reduction of $\simeq-17\,\mathrm{dB}$ is observed at
$V_{\mathrm{p}}=10\,\mathrm{V}$. Fitting the measured curves results in an
estimated $V_{\mathrm{th}}=(151\pm 1)\,\mathrm{mV}$.

We verify the performance of the protocol by directly measuring the squeezing
of the thermal fluctuations $\delta X_{1}(t)$ around its steady-state value
$\bar{X}_{1}$, as quantified by the variance $\sigma_{1}^{\;2}$. We extract
$\sigma_{1}^{\;2}$ from the PSD $S_{X_{1}}(\omega)$, at increasingly larger
parametric drive voltage (Fig. 2(b)). Each resonance peak is then fitted with
a Lorentzian function while discarding the data that lie in the vicinity of 0
Hz. This corresponds to disregarding the coherent amplitude $\bar{X}_{1}$ (and
additional low-frequency noise) and evaluating the variance of the
fluctuations $\delta X_{1}$ only. The resulting fit parameters are used to
calculate the peak area and hence $\sigma_{1}^{\;2}$. The variance as a
function of $V_{\mathrm{p}}$ is shown in Fig. 2(c). Fitting the model in Eq.
(4) returns $V_{\mathrm{th}}=(148\pm 4)\,\mathrm{mV}$.

Despite the strong squeezing which can be achieved through the piezo
actuation, reaching even larger values is experimentally challenging. Applying
larger $V_{\mathrm{p}}\gg 10\,\mathrm{V}$ is not a viable option, due to
increased noise levels of the detected mechanical motion, likely attributable
to heating of the actuator. Nonetheless, on few assemblies we observed an
increase in the squeezing performance—corresponding to a four-fold drop in
$V_{\mathrm{th}}$—when performing the measurement at specific piezo offset
voltage $V_{\mathrm{DC}}$. We attribute this enhancement to the presence of an
in-plane mode with angular frequency
$\Omega_{\mathrm{ip}}(V_{\mathrm{DC}})\approx 2\Omega_{\mathrm{m}}$ (see
Supplementary Information).

### Capacitive parametric drive

Next, we analyze the performance of the capacitor-based design. For this, we
use a $4.8\,\mathrm{mm}\times 5.4\,\mathrm{mm}$ Lotus-class dimer membrane
[Planz23], with a thickness of $50\,\mathrm{nm}$. The dimer symmetric mode has
a resonance frequency $\Omega_{\mathrm{m}}=2\pi\times 1.3\,\mathrm{MHz}$, with
$Q=(0.67\pm 0.04)\times 10^{6}$. We first characterize the frequency shift as
a function of an applied DC voltage. The capacitor has a nonlinear spring-
softening effect (Eq. 5), leading to a change of frequency
$\tilde{\Omega}_{0}\left(V\right)=\Omega_{\mathrm{m}}\sqrt{1-k_{\mathrm{m}}\,C^{{}^{\prime\prime}}(x)|_{x_{\mathrm{eq}}}\,V^{2}/2}$
as shown in the inset of Fig. 3, where $x_{\mathrm{eq}}$ is the equilibrium
position of the membrane at a given bias voltage $V_{\mathrm{DC}}$.

In order to generate squeezing, we apply a bias voltage
$V(t)=V_{\mathrm{DC}}+V_{\mathrm{p}}\cos(\Omega_{\mathrm{p}}t+\varphi)$. The
application of non-zero $V_{\mathrm{DC}}$ is required to reach a low threshold
value, since
$V_{\mathrm{th}}=2k_{\mathrm{m}}/(Q\,V_{\mathrm{DC}}\,C^{{}^{\prime\prime}}(x)|_{x_{\mathrm{eq}}})$
[Bothner2020]. Experimentally, we set the bias voltage to
$V_{\mathrm{DC}}=8\,\mathrm{V}$. Additionally, we observe that applying a bias
voltage increases the quality factor to a value $Q(V)\approx
2\,Q(V_{\mathrm{DC}}=0)$, at the chosen bias voltage.

The variance of $X_{1}$ at different parametric drive amplitudes (Fig. 3) is
obtained following the same steps described in the previous paragraph. Despite
the considerably lower $Q$-factor compared to the resonator used to study the
piezo-actuated drive, the fitted threshold voltage is almost one order of
magnitude less, with a value $V_{\mathrm{th}}=(39\pm 3)\,\mathrm{mV}$. This
results in a maximum observed squeezing of the variance
$(\sigma_{1}/\sigma_{0})^{2}=-21\,\mathrm{dB}$.

Here we remark that the observed value is currently limited by experimental
factors, such as low signal-to-noise ratio, heating due to leakage currents in
the electrodes, and low mechanical quality factor. Simple improvements to the
interferometer sensitivity and to the device fabrication and assembly are
predicted to yield significantly larger squeezing.

Figure 3: Parametric modulation using a capacitive drive. Normalized variance
of the amplitude quadrature $(\sigma_{1}/\sigma_{0})^{2}$ as a function of the
parametric drive voltage, where $\sigma_{0}^{\;2}$ is the variance at
$V_{\mathrm{p}}=0$, using the same protocol described in Fig. 2 and in the
text. The inset shows the frequency tuning of the mode as a function of the
bias voltage $V_{\mathrm{DC}}$. The solid lines are fits to the experimental
data.

##  IV Outlook on quantum squeezing

Figure 4: Prospect for purity and signal-to-noise ratio (SNR) of the
detection. The simulations assume $T=10\,\mathrm{K}$,
$\Omega_{\mathrm{m}}=2\pi\times 1\,\mathrm{MHz}$, $Q=10^{9}$,
$\eta_{\mathrm{det}}=0.77$. (a) Purity of the mechanical state at the optimal
feedback which minimizes $\sigma_{2}^{2}$, as a function of the normalized
squeezing rate and the backaction rate. In the region to the right of the
light-blue-dashed line the expected amplitude standard deviation is below the
quantum zero point fluctuations. (b) Signal-to-noise ratio of the detected
$X_{1}$ quadrature, defined as the ratio of the amplitude of the mechanical
peak ($S_{X_{1}}\left(0\right)$, inset) and the imprecision noise floor
($S_{\mathrm{nn}}=x_{\mathrm{zpf}}^{2}/2\Gamma_{\mathrm{meas}}$), as a
function of the normalized squeezing rate and back action rate. The white line
marks the contour at which $\mathrm{SNR}=1$.

To assess the experimental feasibility of reaching—and observing—quantum
squeezing, we consider a resonant, high finesse optical cavity operating in
the bad cavity regime, characterized by a linewidth
$\kappa/2\pi\gg\Omega_{\mathrm{m}}/2\pi$. Monitoring the output of a
resonantly-driven cavity allows measuring mechanical displacements
characterized by a rate
$\Gamma_{\mathrm{meas}}=\eta_{\mathrm{det}}4g^{2}/\kappa$, where $g$ is the
optomechanical coupling strength [Aspelmeyer2014], $\kappa$ is the cavity
intensity decay rate, $x_{\mathrm{zpf}}=\sqrt{\hbar/(2m\Omega_{\mathrm{m}}})$
and $\eta_{\mathrm{det}}$ is the detection efficiency. The quantum noise of
the probe has two detrimental effects on the performance. Its amplitude
fluctuations lead to the quantum back action force with a (double sided) power
spectral density given by
$S_{\mathrm{qba}}\left(\omega\right)=\hbar^{2}\Gamma_{\mathrm{qba}}/x_{\mathrm{zpf}}^{2}$
where $\Gamma_{\mathrm{qba}}=4g^{2}/\kappa$. The detected signal, referred to
the mechanics, is
$\hat{y}\left(t\right)=\hat{x}\left(t\right)+\hat{x}_{\mathrm{imp}}\left(t\right)$.
The quantum uncertainty in the phase of the probe results in an imprecision
noise floor in the detection ($\delta\hat{y}=\hat{x}_{\mathrm{imp}}$), not
correlated with the mechanics, which the feedback loop attempts to cancel by
actuating the mechanical motion (see Supplementary Information). This results
in a force on the phase quadrature in the Fourier domain:

$\delta\tilde{{F}}_{\mathrm{fb}}\left(\omega\right)=m\Omega_{\mathrm{m}}\Gamma_{\mathrm{fb}}\delta\tilde{y}_{2}\left(\omega\right),$
(6)

where the quadrature of the detected noise signal $\delta\tilde{y}_{2}$ is
defined with the same convention as in Eq. (2).

Adding the backaction of the feedback and of the optical readout in Eq. (3),
the variances of the two quadratures read:

$\displaystyle\sigma_{1}^{2}$
$\displaystyle=\frac{x_{\mathrm{zpf}}^{2}}{1+g_{\mathrm{s}}}\left[\left(2\bar{n}+1\right)+2\gamma_{\mathrm{qba}}\right],$
(7) $\displaystyle\sigma_{2}^{2}$
$\displaystyle=\frac{x_{\mathrm{zpf}}^{2}}{1-g_{\mathrm{s}}+g_{\mathrm{fb}}}\left[\left(2\bar{n}+1\right)+\frac{1}{2}\left(\frac{g_{\mathrm{fb}}^{2}}{4\gamma_{\mathrm{meas}}}+4\gamma_{\mathrm{qba}}\right)\right],$

where $\gamma_{\mathrm{meas}}=\eta_{\mathrm{det}}\gamma_{\mathrm{qba}}$,
$\gamma_{\mathrm{qba}}=\Gamma_{\mathrm{qba}}/\Gamma_{\mathrm{m}}$ and
$\bar{n}$ is the average occupancy of the thermal bath.

Through Eq. (7), we can estimate the expected purity of the state as a
function of the quantum back action and the parametric gain (Fig. 4(a)) for
realistic device parameters [Rossi_2018]. For each value of
$\gamma_{\mathrm{qba}}$ and $g_{\mathrm{s}}$, the feedback gain is tuned to
minimize $\sigma_{2}^{2}$. The purity approaches $\sqrt{\eta_{\mathrm{det}}}$
in the limit of strong squeezing and fast measurement rate.

The expected detection signal-to-noise ratio (SNR) for the amplitude
quadrature is reported in Fig. 4(b). In the limit of strong measurement rate,
where the mechanical motion is dominated by the quantum back action, it is
possible to squeeze the amplitude quadrature below $x_{\mathrm{zpf}}$ and
detect it with a unity signal-to-noise ratio (SNR) above the imprecision
noise.

##  V Conclusions

By applying active stabilization of the anti-squeezed quadrature, we have
shown that both piezo- and capacitively-actuated parametric driving can be
used to squeeze the mechanical motion far beyond the $3\,\mathrm{dB}$ limit,
reaching $17\,\mathrm{dB}$ and $21\,\mathrm{dB}$, respectively. Improvements
of the device design in the case of piezo actuation are not expected to vastly
improve the maximum parametric gain. In contrast, we envision that in the case
of capacitive drive, realistic improvements—in terms of larger mechanical
quality factor, smaller membrane-electrode separation, and reduced leakage
current within the circuit—will result in much larger attainable squeezing,
with a scaling of $\sigma_{1}^{2}\propto
d_{0}^{3}/V_{\mathrm{DC}}V_{\mathrm{p}}Q$ for $g_{\mathrm{s}}\gg 1$. Under
such modifications, the device will remain compatible with force sensing
applications [Halg2021].

We also provide a theoretical model to predict the behaviour of the system in
the quantum regime and to discuss the feasibility of squeezing below the zero-
point motion. At a moderate bath temperature of $10\,\mathrm{K}$, quantum
squeezing requires thermomechanical squeezing by more than $56\,\mathrm{dB}$.
We find this can be reached with conservative device parameters
($d_{0}=1\,\mathrm{\mu m}$, $m_{\mathrm{eff}}=30\,\mathrm{ng}$,
$\Omega_{\mathrm{m}}=2\pi\times\,1\,\mathrm{MHz}$ and $Q=10^{9}$) with
voltages below the threshold for (e.g., pull-in) instability.

##  VI Acknowledgements

We thank Emil Zeuthen and Jan Košata for fruitful discussions. This work was
supported by the European Research Council project PHOQS (Grant No.
101002179), the Novo Nordisk Foundation (Grant No. NNF20OC0061866), as well as
the Independent Research Fund Denmark (Grant No. 1026-00345B). L. S.
acknowledges financial support from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No
101063285.

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