# The limit points of the bass notes of arithmetic hyperbolic surfaces

Michael Magee
_To Peter Sarnak on the occasion of his 70th birthday_

###### Abstract

We prove that the limit points of the bass notes of arithmetic hyperbolic
surfaces are the interval $\left[0,\frac{1}{4}\right]$.

###### Contents

  1. 1 Introduction
    1. 1.1 Related works
    2. 1.2 Acknowledgments
  2. 2 Set up
  3. 3 Probabilistic input
  4. 4 Probabilistic spectral gap
  5. 5 Geometric tangle freeness
  6. 6 Local bounds for eigensections
  7. 7 Bumping off the cusps
  8. 8 Bumping to zero along tile edge
  9. 9 Proof of Theorems 1.2 and 1.1
  10. A Common fixed points of subgroups under random homomorphisms to permutation groups

##  1 Introduction

######  Theorem 1.1.

Let

$\mathsf{BASS}(\mathsf{arith}_{2})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{\,\lambda_{1}(\Lambda\backslash\mathbb{H})\,:\,\Lambda\text{
is an arithmetic lattice in $\mathrm{PSL}_{2}(\mathbb{R})\,\\}.$}$

The limit points of $\mathsf{BASS}(\mathsf{arith}_{2})$ are the interval
$\left[0,\frac{1}{4}\right]$. In fact this is also true if we restrict to
$\Lambda\leq\mathrm{PSL}_{2}(\mathbf{Z})$.

Here $\lambda_{1}(Y)$ is the bottom of the non-trivial111A multiplicity-one
zero is removed. spectrum of the Laplace–Beltrami operator on $Y$.

This theorem confirms a prediction of Sarnak that is part of his program
‘Prescribing the spectra of locally uniform geometries’. This program was
described during his Chern lectures [Sar23] (U.C. Berkeley, Jan.-Feb. 2023)
and encompasses, among other things, the geometry of numbers, the Markoff
spectrum, the Oppenheim conjecture, and results about Euclidean sphere
packings in one viewpoint. Let us now point to the backdrop of Theorem 1.1.

0. The fact that $0$ is a limit point of $\mathsf{BASS}(\mathsf{arith}_{2})$ follows from an old result due to Randol [Ran74] and independently, Selberg [Sel65].

1. Selberg’s Eigenvalue Conjecture [Sel65] says that $\frac{1}{4}$ is the only possible value of $\lambda_{1}(\Lambda\backslash\mathbb{H})$ when $\Lambda$ is a congruence subgroup of $\mathrm{PSL}_{2}(\mathbf{Z})$. The best progress towards this conjecture is Kim–Sarnak [Kim03, App. 2] giving a lower bound of $\frac{975}{4096}$.

2. There are obviously only finitely many subgroups of $\mathrm{SL}_{2}(\mathbf{Z})$ with a given upper bound on their index and hence covolume in $\mathbb{H}$. Hence in establishing Theorem 1.1 one must have

$\mathrm{vol}\left(\Lambda_{i}\backslash\mathbb{H}\right)\to\infty.$

If one extends consideration to all arithmetic lattices in
$\mathrm{PSL}_{2}(\mathbb{R})$ the same is true by a result of Borel [Bor81].

3. It is clear that for non-compact lattices in $\mathrm{PSL}_{2}(\mathbb{R})$, $\lambda_{1}(\Lambda\backslash\mathbb{H})\in\left[0,\frac{1}{4}\right]$ since there is continuous spectrum of the Laplacian with support $\big{[}\frac{1}{4},\infty\big{)}$ [LP81]. For cocompact arithmetic lattices, a result of Huber [Hub74] states $\limsup\lambda_{1}(\Lambda\backslash\mathbb{H})\leq\frac{1}{4}$ for any sequence with covolume tending to infinity; combined with the aforementioned result of Borel, this means any limit point of $\lambda_{1}$ must be in $\left[0,\frac{1}{4}\right]$.

4. The fact that $\frac{1}{4}$ is a limit point of $\mathsf{BASS}(\mathsf{arith}_{2})$ follows from work of Hide and the author [HM23]. If $\mathsf{BASS(arith_{2}^{c})}$ denotes the bass notes of compact arithmetic surfaces then $\frac{1}{4}$ is a limit point of $\mathsf{BASS(arith_{2}^{c})}$ by work of Hide, Louder, and the author [LM23]. It seems possible to combine the methods of the current paper with estimates towards the Selberg conjecture [Kim03] (and Jacquet–Langlands [JL70]) to prove the limit points of $\mathsf{BASS(arith_{2}^{c})}$ contain the interval $\left[0,\frac{975}{4096}\right]$. There is somewhat of a barrier222It seems to require knowing something like this. For $\Gamma_{2}$ a surface group of genus 2, do there exist a sequence of $\phi_{i}:\Gamma_{2}\to S_{n_{i}}$ which are tangle-free on scale $c(\log n_{i})^{\alpha}$, $c,\alpha>0$, and such that $\mathrm{std}\circ\phi_{i}:\Gamma_{2}\to\mathcal{U}(n_{i}-1)$ strongly converge to the regular representation of $\Gamma_{2}$? to use strong convergence as in this paper to obtain the maximal interval $[0,\frac{1}{4}]$ as limit points of $\mathsf{BASS(arith_{2}^{c})}$, this therefore remains a very interesting open problem.

The result corresponding to Theorem 1.1 for $d$-regular graphs, $d\geq 3$
fixed, in place of arithmetic hyperbolic surfaces has recently been proved by
Alon and Wei [AW23]. See $\S\S$1.1 for a more detailed comparison between
these works.

On route to Theorem 1.1 we prove a theorem that may be of independent
interest. Before stating it we recall some notation. Given a finite set $S$
and two functions $\xi_{1},\xi_{2}:S\to\\{1,-1\\}$, the _Hamming distance_
$d_{\mathsf{Ham}}(\xi_{1},\xi_{2})$ between $\xi_{1}$ and $\xi_{2}$ is by
definition the number of points in $S$ at which the functions disagree.

Given a finite-area hyperbolic surface $Y=\Gamma\backslash\mathbb{H}$, let

$\mathsf{Jac}^{(2)}(Y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathrm{Hom}(\Gamma,\mathbf{Z}/2\mathbf{Z})$

be the 2-torsion in the Jacobian variety of $Y$. Given any
$\chi\in\mathsf{Jac}^{(2)}(Y)$ there is an Hermitian line bundle
$\mathcal{L}_{\chi}$ over $Y$ associated to $\chi$ and with associated
Laplacian operator etc; let $\lambda_{1}(\mathcal{L}_{\chi})$ denote the
smallest eigenvalue of this operator, if it exists, or $\frac{1}{4}$
otherwise. Given a finite generating set $\mathcal{B}$ of $\Gamma$, and
$\chi_{1},\chi_{2}\in\mathsf{Jac}^{(2)}(Y)$ let

$\displaystyle d_{\mathsf{Ham}}^{\mathcal{B}}(\chi_{1},\chi_{2})$
$\displaystyle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}d_{\mathsf{Ham}}\text{(}(\chi_{1}(b):b\in\mathcal{B}),(\chi_{2}(b):b\in\mathcal{B})).$

Our main technical contribution is the following.

######  Theorem 1.2.

Let $X=\Gamma(2)\backslash\mathbb{H}$. There is $c>0$ such that the following
holds. For any $\lambda_{0}<\frac{1}{4}$, there is $C(\lambda_{0})>0$ such
that if $Y_{n}$ is a uniformly random degree $n$ cover of $Y$ then with
probability tending to one as $n\to\infty$ there is a generating set
$\mathcal{B}_{n}$ of $\pi_{1}(Y_{n})$ such that if
$d_{\mathsf{Ham}}^{\mathcal{B}_{n}}(\chi_{1},\chi_{2})=1$ and
$\lambda_{1}(\mathcal{L}_{\chi})\leq\lambda_{0}$ then

$|\lambda_{1}(\mathcal{L}_{\chi_{1}})-\lambda_{1}(\mathcal{L}_{\chi_{2}})|<C(\lambda_{0})n^{-c\sqrt{\frac{1}{4}-\lambda}}.$

Remark While stated as a probabilistic result, the proof relies only on two
deterministic conditions: that $Y_{n}$ is connected, and that $Y_{n}$
satisfies GTF($K\log n$) of $\S$3 for some uniform $K>0$ not depending on $n$.

We prove that GTF($K\log n)$ holds for random covers $Y_{n}$ as above for some
uniform $K>0$. This is a geometric version of the tangle free condition used
by Friedman [Fri08]. A geometric version of this property was proven to hold
for compact Weil–Petersson random surfaces by Monk and Thomas [MT22] (see also
[Mon20, GLMST21]). A very recent and independent preprint of Klukowski and
Marković [KM24] proves a related geometric property — the ‘$L$-horoball
property’ — of random covers of a cusped hyperbolic surface.

###  1.1 Related works

Graphs. As we mentioned before, the analog of Theorem 1.1 for all $d$-regular
graphs was proved by Alon and Wei [AW23].

Our approach certainly has the same overall strategy as Alon–Wei (a type of
discrete continuity between two known end points) but differs in several ways:

  * •

We use two–covers rather than flipping edges in graphs. While it is possible
flipping edges could have worked in our setting, using two-covers provides an
alternative method that possibly extends more easily to more general manifolds
e.g. hyperbolic three-manifolds. We plan to return to this topic.

  * •

We produce (doubly) exponentially stronger bounds on the type of continuity
one can obtain. This is at least partly because we relax girth conditions to
tangle-free conditions, but are able to use them in a similar way.

  * •

Obviously, we have to deal with all complications arising from the geometric
setting, with the especial complication that our surfaces are not compact
(this complication was chosen in favor of having non-free fundamental groups,
which is a separate issue in the compact case).

In the setting of $d$-regular graphs, one could also ask about the bass notes
of only arithmetic __ graphs (those that arise as locally symmetric spaces
attached to arithmetic lattices in e.g. $\mathrm{GL_{2}}(\mathbb{Q}_{p}))$.
This can likely be dealt with by covering spaces as we do here. In this case
it may also be useful to use two-covers as in this paper instead of flips.

Surfaces. In the context of (compact) hyperbolic orbifolds, there is the
analog question of Theorem 1.1 for all compact hyperbolic orbifolds. This has
been studied by Kravchuk, Mazac, and Pal [KMP21] who obtain rigorous bounds on
Laplacian spectra via a framework inspired by the conformal bootstrap in
conformal field theory in conjunction with linear programming. This allows
them to formulate a precise conjecture: if

$\mathsf{BASS}(\mathsf{hyp}_{2}^{c})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{\,\lambda_{1}(\Lambda\backslash\mathbb{H})\,:\,\Lambda\text{
a cocompact lattice in $\mathrm{PSL}_{2}(\mathbb{R})\,\\}$},$

then the limit points of $\mathsf{BASS}(\mathsf{hyp}_{2}^{c})$ should be an
interval

$[0,15.79023...]$

where the right hand endpoint is $\lambda_{1}$ of a specific hyperbolic
orbifold. In _(ibid.)_ this conjecture is reduced [KMP21, Thm. 4.3] to four
inequalities concerning $\lambda_{1}$ of specific orbifolds (that should be
numerically check-able) plus a conjecture about the global maximum of
$\lambda_{1}$ on the moduli space of genus $0$ orbifolds with orbifold points
of orders $(2,2,2,3)$.

Discrete bass note spectra. While the current work addresses the _limit
points_ of $\mathsf{BASS}(\mathsf{arith}_{2})$, and completes the picture for
non-compact surfaces333At least, without beginning a discussion about cusp
forms., in the case of compact surfaces the _discrete_ points of
$\mathsf{BASS}(\mathsf{arith}_{2}^{c})$ and
$\mathsf{BASS}(\mathsf{hyp}_{2}^{c})$ are also of basic interest. Sarnak
conjectures [Sar23] that there are infinitely many such discrete points of
$\mathsf{BASS}(\mathsf{arith}_{2}^{c})$: concretely, there are infinitely many
values $\lambda=\lambda_{1}(Y)>\frac{1}{4}$ with $Y$ arithmetic compact. This
remains open.

In the setting of $\mathsf{BASS}(\mathsf{hyp}_{2}^{c})$, in contrast to the
arithmetic setting, Kravchuk, Mazac, and Pal [KMP21] explain that there should
be three (precisely known) discrete points of
$\mathsf{BASS}(\mathsf{hyp}_{2}^{c})$ and this is proved modulo resolving the
same issues as previously discussed in the context of _(ibid.)._

###  1.2 Acknowledgments

We thank Noga Alon, Charles Bordenave, Benoit Collins, Vlad Marković, Doron
Puder, Peter Sarnak, and Joe Thomas for conversations about this work.

Funding: This material is based upon work supported by the National Science
Foundation under Grant No. DMS-1926686. This project has received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 949143).

### Notation

If $S$ is a finite set we write $\ell_{0}^{2}(S)$ for the set of $\ell^{2}$
functions on $S$ that are orthogonal to constant functions. We use the
notation $[n]\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{1,\ldots,n\\}$ for
$n\in\mathbb{N}$.

##  2 Set up

Let
$\Gamma\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma(2)\subset\mathrm{PSL}_{2}(\mathbf{Z})$
be the (projective image of the) principal congruence subgroup of level 2.
This $\Gamma$ is free of rank 2 on the generators

$a\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(\begin{array}[]{cc}1&2\\\
0&1\end{array}\right),\quad
b\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(\begin{array}[]{cc}1&0\\\
2&1\end{array}\right),$

and $X\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma\backslash\mathbb{H}$ is
a hyperbolic 3-times punctured sphere.

Figure 2.1: $F$, fundamental domain for $\Gamma$

We use $\phi$ to denote a homomorphism $\phi:\Gamma\to S_{n}$. Given $\phi$,
$\Gamma$ acts on $\mathbb{H}\times[n]$ by
$\gamma(z,x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(\gamma
z,\phi(\gamma)x)$. We denote by

$X_{\phi}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma\backslash_{\phi}\left(\mathbb{H}\times[n]\right)$

the quotient by this action. The resulting $X_{\phi}$ is a degree $n$
Riemannian covering space of $X$ and as such a hyperbolic surface. If
$X_{\phi}$ is connected — later we will ensure this — then

$X_{\phi}\cong\Gamma_{\phi}\backslash\mathbb{H}$

where $\Gamma_{\phi}$ is the stabilizer subgroup

$\Gamma_{\phi}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathrm{Stab}_{\phi}(1).$

Let

$\mu_{2}=\mu_{2}(\mathbb{C})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(\\{\pm
1\\},\times)$

be the multiplicative group of the complex square roots of unity.

Now consider a general homomorphism $\chi:\Gamma_{\phi}\to\mu_{2}$. If $\chi$
is non-trivial, it defines a (Galois) double cover $X_{\phi,\chi}$ of
$X_{\phi}$ via

$X_{\phi,\chi}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\ker(\chi)\backslash\mathbb{H}.$

Smooth functions on $X_{\phi,\chi}$ that are odd under the deck transformation
involution correspond to smooth sections of the Hermitian line bundle

$\mathbb{C}\to\mathcal{L}_{\chi}\to X_{\phi}$

associated to $\chi$. This correspondence intertwines the following table of
differential operators (acting on local trivializations of sections).

$C_{\mathrm{odd}}^{\infty}(X_{\phi,\chi})$ | $C^{\infty}(X_{\phi};\mathcal{L}_{\chi})$
---|---
$\nabla_{X_{\phi,\chi}}$ | $\nabla_{X_{\phi}}$
$\Delta_{X_{\phi,\chi}}$ | $\Delta_{X_{\phi}}$

Let $F$ be an ideal hyperbolic quadrilateral in $\mathbb{H}$ with vertices at
$-1,0,1,\infty$. Then $F$ is a fundamental domain for $\Gamma(2)$ with side
pairings given by $a,b$ as depicted in Figure 2.1. Let $K$ denote the bounded
component of $X$ obtained by cutting along the three closed horocycles444By
the collar lemma for non-compact surfaces [Bus92, Lemma 4.4.6], these
horocycles are disjoint.  of length one. Let $\mathcal{K}\subset F$ denote the
subset corresponding to $K$. Let $K_{\phi}$ denote the preimage of the core
$K$ in $X_{\phi}$. This is bounded by some closed horocycles that are lifts of
the horocycles bounding $K$. This $K_{\phi}$ is tiled by copies of
$\mathcal{K}$.

The complement of $K_{\phi}$ in $X_{\phi}$ is a union of disjoint cusps. For
$z\in X_{\phi}\backslash K_{\phi}$, let $y(z)$ denote the coordinate of

$z=x+iy(z)$

in a model of the universal cover of the cusp in the hyperbolic upper half
plane where the bounding horocycle of the cusp is covered by the line
$\mathrm{Im}(z)\equiv 1$. For example, if $y(z)=1$, then $z\in\partial
K_{\phi}$. We call this the _standard model of the cusp._

##  3 Probabilistic input

We say that a family of events hold asymptotically almost surely (a.a.s.) if
they hold with probability tending to one as $n\to\infty$. We will prove that
for uniformly random

$\phi\in\mathrm{Hom}(\Gamma,S_{n})$

the following simultaneously happen a.a.s.

Connected

$X_{\phi}$ is connected. The fact that $X_{\phi}$ is connected a.a.s. follows
from a result of Dixon [Dix69] that states that two uniformly random and
independent elements of $S_{n}$ generate $A_{n}$ a.a.s. This implies that the
image of $\phi$ is a.a.s. transitive on $[n]$, and this is what is required to
ensure $X_{\phi}$ connected.

Spectral gap

For any non-trivial $\theta\in\mathrm{Hom}(\Gamma,\mu_{2})$, let
$\chi(\theta)$ denote the restriction of $\theta$ to $\Gamma_{\phi}$. We have
a.a.s., for any non-trivial such $\theta$

$\lambda_{1}(X_{\phi,\chi(\theta)})\geq\frac{1}{4}-o_{n\to\infty}(1).$ (3.1)

Remark. This relies crucially on the special fact that any connected degree
two covering space $X_{\theta}$ of $X$ has
$\lambda_{1}(X_{\theta})\geq\frac{1}{4}$. This, as well as (3.1) will be
established in $\S$4.

Geometric tangle freeness

GTF($c\log n)$ holds for some $c>0$ not depending on $n$:
GTF($R$): For any $z\in X_{\phi}$, the fundamental group
$\pi_{1}(B_{X_{\phi}}(z,R))$ is cyclic (possibly trivial).
This will be established in $\S$5.

Since there only finitely many conditions above, it obviously suffices to
prove each occur a.a.s.

##  4 Probabilistic spectral gap

We assume $X_{\phi}$ is connected here since we already explained this holds
a.a.s.

First we establish the needed spectral properties of degree two covers of $X$.
There is a one-to-one correspondence between non-trivial
$\theta\in\mathrm{Hom}(\Gamma,\mu_{2})$ and connected degree two covers
$X_{\theta}$ of $X$ via taking kernels. Any non-trivial such $\theta$ factors
as

$\Gamma\to\Gamma/[\Gamma,\Gamma]\to
H_{1}(\Gamma,\mu_{2})\cong\mu_{2}\times\mu_{2}\to\mu_{2}$

where the first two maps are canonical. Consider $\Gamma(4)$, the subgroup of
$\Gamma$ consisting of elements of $\Gamma$ congruent to the class of $\pm I$
modulo 4. It is not hard to check that the quotient

$\Gamma\to\Gamma/\Gamma(4)$

agrees with the map $\Gamma\to H_{1}(\Gamma,\mu_{2})$ above and hence any
connected degree two cover $X_{\theta}$ of $X$ is an intermediate cover

$X(4)=\Gamma(4)\backslash\mathbb{H}\to X_{\theta}\to\mathbb{H}.$

By a result of Huxley [Hux85, §6], $\lambda_{1}(X(4))\geq\frac{1}{4}$. This
implies $\lambda_{1}(X_{\theta})\geq\frac{1}{4}$.

Now take an arbitrary non-trivial $\theta\in\mathrm{Hom}(\Gamma,\mu_{2})$ and
let $\chi(\theta)$ be the restriction of $\theta$ to $\Gamma_{\phi}$. Hide and
the author [HM23, Thm. 1] proved that since $\lambda_{1}(X)\geq\frac{1}{4}$
(by Huxley [Hux85] again), a.a.s.

$\lambda_{1}(X_{\phi})\geq\frac{1}{4}-o_{n\to\infty}(1).$

We explain how to generalize this result to obtain the same for
$X_{\phi,\chi(\theta)}$. As we explained in $\S$2, odd functions on
$X_{\phi,\chi(\theta)}$ correspond to sections of $\mathcal{L}_{\chi(\theta)}$
in a Laplacian-respecting way. Sections of $\mathcal{L}_{\chi(\theta)}$
correspond to sections of the Hermitian vector bundle

$V_{\theta\otimes\rho_{\phi}}\to X$

associated to the unitary representation $\theta\otimes\rho_{\phi}$, where
$\rho_{\phi}$ is the composition

$\rho_{\phi}:\Gamma\xrightarrow{\phi}\text{$\mathrm{Perms}$}([n])\to\mathcal{U}(\ell^{2}([n])).$

Recall that if $S$ is a finite set, $\ell_{0}^{2}(S)$ denotes the
orthocomplement in $\ell^{2}(S)$ of the constant functions with respect to the
standard inner product. We have a splitting
$\rho_{\phi}\cong\mathbf{1}\oplus\rho_{\phi}^{0}$ where

$\rho_{\phi}^{0}:\Gamma\xrightarrow{\phi}\text{$\mathrm{Perms}$}([n])\to\mathcal{U}(\ell_{0}^{2}([n])).$

This means

$\theta\otimes\rho_{\phi}\cong\theta\oplus\left(\theta\otimes\rho_{\phi}^{0}\right).$

Sections of the associated bundle to the $\theta$ direct summand above
correspond exactly to odd functions on $X_{\phi,\chi(\theta)}$ that are lifts
from odd functions on $X_{\theta}$. This only yields eigenvalues
$\geq\frac{1}{4}$ by previous remarks on the spectrum of $X_{\theta}$. Thus
the only sections of $\mathcal{L}_{\chi(\theta)}$ that can correspond to
eigenvalues of the Laplacian below $\frac{1}{4}$ correspond to sections of the
sub-Hermitian vector bundle over $X$ associated to
$\theta\otimes\rho_{\phi}^{0}$. We now use two facts from operator algebras.

A. If $\rho_{\phi}^{0}$ converge strongly in probability to the regular
representation $\lambda_{\Gamma}$ of $\Gamma$ (as they do in the current case
by the main result of Bordenave–Collins [BC19]) then for any finite
dimensional representation $\theta,$ $\theta\otimes\rho_{\phi}^{0}$ converge
strongly in probability to

$\theta\otimes\lambda_{\Gamma}.$

This is known as ‘matrix amplification’ e.g. [HT05, §9].

B. On the other hand, by Fell’s absorption principle,
$\theta\otimes\lambda_{\Gamma}\cong\lambda_{\Gamma}$. So the upshot is that in
this case, $\theta\otimes\rho_{\phi}^{0}$ converges strongly to
$\lambda_{\Gamma}$.

Now the method of [HM23], mutatis mutandis, implies that if
$\lambda_{1}(\mathcal{L}_{\chi(\theta)})$ denotes the bottom of the spectrum
of the Laplacian on sections, then a.a.s.

$\lambda_{1}(\mathcal{L}_{\chi(\theta)})\geq\frac{1}{4}-o_{n\to\infty}(1).$

By the previous discussion, this gives a.a.s.

$\lambda_{1}(X_{\phi,\chi(\theta)})\geq\frac{1}{4}-o_{n\to\infty}(1),$

i.e. Spectral gap holds a.a.s.

##  5 Geometric tangle freeness

Let $\epsilon>0$ be a constant to be chosen later on. Recall for $z\in
X_{\phi}\backslash K_{\phi}$ the notation $y(z)$. If $y(z)\geq n^{\epsilon}$
then the distance from $y$ to $\partial K_{\phi}$ is at least

$d(y,\partial K_{\phi})\geq\epsilon\log n.$

This means that $B_{X_{\phi}}(y,\epsilon\log n)$ has at most cyclic $\pi_{1}$
and hence GTF($\epsilon\log n$) holds. So we may assume $y(z)<n^{\epsilon}$
henceforth — our constant $c$ will be at most whatever $\epsilon$ we choose.

Let $K_{\phi}(\epsilon\log n)$ (resp. $K(\epsilon\log n)$) be the points of
$X_{\phi}$ (resp. $X$) at distance less than $\epsilon\log n$ from $K_{\phi}$
(resp. $K$). So we currently assume $z\in K_{\phi}(\epsilon\log n)$. Fix some
$o\in K$. Since $K$ is compact there is some $R_{0}>0$ such that $K\subset
B_{X}(o,R_{0})$, hence

$K(\epsilon\log n)\subset B_{X}(o,R_{0}+\epsilon\log n).$ (5.1)

The fiber of $o$ under the covering map has representatives

$\displaystyle[(o,i)]$
$\displaystyle\in\Gamma\backslash_{\phi}\left(\mathbb{H}\times[n]\right),\quad
i\in[n].$

Suppose that $z\in K_{\phi}(\epsilon\log n)$, which is tiled by copies of
$K(\epsilon\log n)$. Let $i$ be such that $[(o,i)]$ is in the same
$K(\epsilon\log n)$-tile as $z$. Then from (5.1),

$z\in B_{X_{\phi}}([(o,i)],R_{0}+\epsilon\log n).$

Let $R>0$ be a parameter. If $B_{X_{\phi}}(x,R)$ has larger than cyclic
$\pi_{1}$, there are two simple non-trivial geodesic loops beginning and
ending at $x$ — not necessarily smoothly joining at $x$ — of length $\leq 4R$.
This means there are two simple non-trivial geodesic loops beginning and
ending at $[(o,i)]$ of length $\leq 4R+2R_{0}+2\epsilon\log n$.

Any such simple geodesic loop covers a simple geodesic loop in $X$ beginning
and ending at $o$. This loop represents an element $\gamma\in\pi_{1}(X,o)$. It
follows that $i$ is a fixed point of $\phi(\gamma)$ and

$d(o,\gamma o)\leq 4R+2R_{0}+2\epsilon\log n.$

In the situation of the previous paragraph, we obtained:

  * •

a number $i\in[n]$

  * •

two non-identity elements $\gamma_{1},\gamma_{2}\in\Gamma$ that generate a
free rank two group $H\leq\Gamma=\pi_{1}(X,o)$, such that $i$ is a common
fixed point of $\phi(H)$ and $d(o,\gamma_{i}o)\leq 4R+2R_{0}+\epsilon\log n$.

We will now show that the above a.a.s. never happens: note that we have
majorized the event of geometric tangles by an event depending only on
elements of $\Gamma$ and $\phi$.

Lattice point count

Consider that $K$ has finite diameter $D$ and volume $V$. If $\gamma o\in
B_{\mathbb{H}}(o,r)$ for $\gamma\in\Gamma$ then $\gamma\mathcal{K}\subseteq
B_{\mathbb{H}}(o,r+D)$, and since $B_{\mathbb{H}}(o,r+D)$ has volume $\asymp
e^{r+D}\asymp e^{r}$, there are at most $\asymp V^{-1}e^{r}\asymp e^{r}$ such
$\gamma$. In summary,

$\left|\left\\{\,\gamma\in\Gamma\,:\,d(o,\gamma o)\leq r\,\right\\}\right|\ll
e^{r}.$ (5.2)

Word lengths

Let $\gamma\in\Gamma$ such that $d(o,\gamma o)\leq r$. We wish to bound the
word length of $\gamma$ with respect to the generators $a,b$, or what is the
same, the number of times the geodesic arc in $\mathbb{H}$ from $o$ to $\gamma
o$ cuts a side of the tiling of $\mathbb{H}$ given by $\Gamma$-translates of
$F$.

Fix $\gamma$ as above and let $\alpha$ denote this aforementioned geodesic
arc. We subdivide $\alpha$ according to times when it leaves and enters

$\mathfrak{K}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bigcup_{\gamma^{\prime}\in\Gamma}\gamma^{\prime}\overline{\mathcal{K}}.$

By perturbing $o$ an infinitesimal amount we can assume $\alpha$ does not
intersect any ‘corner’ in
$\partial\mathfrak{K}\cap\bigcup_{\gamma^{\prime}\in\Gamma}\gamma^{\prime}\partial
F$ — this is not really an important point, only a technical one.

Any thus obtained subsegment $\alpha^{\prime}$ of $\alpha$ that is _outside_
$\mathfrak{K}$ intersects
$\bigcup_{\gamma^{\prime}\in\Gamma}\gamma^{\prime}\partial F$
$\leq\exp(2\ell_{\mathbb{H}}(\alpha^{\prime}))$ times by direct calculation in
the hyperbolic metric.

Any subsegment $\alpha^{\prime}$ that is _inside_ $\mathfrak{K}$ intersects
$\bigcup_{\gamma^{\prime}\in\Gamma}\gamma^{\prime}\partial F$ at most
$1+C\ell_{\mathbb{H}}(\alpha^{\prime})\leq\exp(C\ell_{\mathbb{H}}(\alpha^{\prime}))$
times for some $C\geq 2$. In total then the number of intersections is

$\leq\sum_{\alpha^{\prime}}\exp(C\ell_{\mathbb{H}}(\alpha^{\prime}))\leq\prod_{\alpha^{\prime}}\exp(C\ell_{\mathbb{H}}(\alpha^{\prime}))=\exp(C\ell_{\mathbb{H}}(\alpha)).$

In summary, there is $C\geq 2$ such that $d(o,\gamma o)\leq r$ implies that
the word length of $\gamma$ is at most $e^{Cr}$.

Probabilities of common fixed points

Now fix $\epsilon>0$ such that

$6C\epsilon,2\epsilon<\frac{1}{4}.$ (5.3)

Let $\mathcal{H}(R,n)$ denote the collection of rank two subgroups of $\Gamma$
generated by two elements $\gamma_{1}$ and $\gamma_{2}$ with
$d(o,\gamma_{i}o)\leq 4R+2R_{0}+\epsilon\log n$. Previously we proved
GTF($R$)__ was majorized by the event that some element of $\mathcal{H}(R,n)$
has a common fixed point in $[n]$ under $\phi$. Let $\mathrm{fix}_{H}(\phi)$
denote the number of common fixed points of $\phi(H)$ and suppose

$n\geq e^{6C(4R+2R_{0})}.$

Then Proposition A.1 implies that for any $H\in\mathcal{H}(R,n)$

$\mathbb{E}_{n}[\mathrm{fix}_{H}]\leq\frac{e^{6C(4R+2R_{0}+\epsilon\log
n)}}{n}\stackrel{{\scriptstyle\eqref{eq:epsilon1}}}{{\ll}}\frac{n^{\frac{1}{4}}e^{C^{\prime}R}}{n}$

for some $C^{\prime}>0$. Then

$\displaystyle\mathbb{E}_{n}[\\#\text{ common fixed points of subgroups $H$
for $H\in\mathcal{H}(R)]$}$
$\displaystyle\leq\sum_{H\in\mathcal{H}(R)}\mathbb{E}_{n}[\mathrm{fix}_{H}]$
$\displaystyle\ll\sum_{H\in\mathcal{H}(R)}\frac{e^{C^{\prime}R}}{n^{\frac{3}{4}}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:lattice_point_count}}}{{\ll}}e^{2(4R+2R_{0}+\epsilon\log
n)}\frac{e^{C^{\prime}R}}{n^{\frac{3}{4}}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:epsilon1}}}{{\ll}}\frac{e^{C^{\prime\prime}R}}{n^{\frac{1}{2}}}$

for some $C^{\prime\prime}>0$. Now take

$R=\frac{1}{4C^{\prime\prime}}\log n$

to obtain

$\mathbb{E}_{n}[\\#\text{ common fixed points of subgroups $H$ for
$H\in\mathcal{H}(R,n)]$}\ll n^{-\frac{1}{4}}$

so by Markoff’s inequality

$\mathbb{P}_{n}[\text{there is any $H\in\mathcal{H}(R,n)$ with any common
fixed point]$\ll n^{-\frac{1}{4}}$. }$

Hence a.a.s. GTF($c\log n$) holds if
$c=\min\left(\frac{1}{4C^{\prime\prime}},\epsilon\right)>0$.

##  6 Local bounds for eigensections

Recall from $\S$3 GTF($R$): For any $z\in X_{\phi}$, the geometric ball
$B_{X_{\phi}}(z,R)$ has cyclic fundamental group.

In this section, let $\phi\in\mathrm{Hom}(\Gamma,S_{n})$, assume $X_{\phi}$ is
connected, and let $\chi\in\mathrm{Hom}(\Gamma_{\phi},\mu_{2})$. The purpose
of this section is given an eigensection $f$ of $\mathcal{L}_{\chi}$, with
eigenvalue $\lambda$, to give a local bound on $|f|$ that only depends on
$\lambda$ and $\text{$\phi$},$ under the assumption GTF($c\log n$) for $c$ as
in $\S$3.

######  Proposition 6.1.

For any $0<c^{\prime}<c$ the following holds. Let $\phi,\chi$ be as above.
Assume that $\phi$ satisfies Connected and GTF($c\log n$). Suppose that $f$ is
any smooth section of $\mathcal{L}_{\chi}$ with eigenvalue
$\lambda\in[0,\frac{1}{4})$ and $\|f\|_{L^{2}(X_{\phi})}=1$. Then:

  1. 1.

If $z\in X_{\phi}\backslash K_{\phi}$, and $L$ is the length of the closed
horocycle bounding the cusp containing $z$, then

$|f(z)|^{2}\ll_{c}n^{-c\sqrt{\frac{1}{4}-\lambda}}\left(1+\frac{y(z)}{L}\log\left(\frac{2y(z)n^{c}}{L}\right)\right).$

  2. 2.

If $z\in K_{\phi}$ then

$|f(z)|^{2}\ll_{c^{\prime},c}n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$

The implied constants depend on $\Gamma$ and $c^{\prime},c$ but nothing else.

We will adapt and extend the argument of Gilmore–Le Masson–Sahlsten–Thomas
[GLMST21] to the present context of cusped surfaces and also explain why it is
uniform over line bundles. We will apply the pre-trace inequality from [Gam02,
Prop. 5.2] that we recall here for the sake of the reader.

######  Lemma 6.2.

For $\lambda\in[0,\frac{1}{4}]$ let
$r(\lambda)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}i\sqrt{\frac{1}{4}-\lambda}$.
Let
$0=\lambda_{0}\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{q}\leq\frac{1}{4}$
denote the eigenvalues of the Laplacian on sections of $\mathcal{L}_{\chi}$
that are at most $\frac{1}{4}$, let $f_{j}$ denote their corresponding
$L^{2}$-normalized eigensections, viewed as $\chi$-equivariant functions for
$\Gamma_{\phi}$ acting on $\mathbb{H}$. For any compactly supported
$k:[0,\infty)\to\mathbb{R}$, with non-negative Selberg transform $h$, for any
$z\in\mathbb{H}$

$\sum_{j\in[q]}h(r(\lambda_{j}))|f_{j}(z)|^{2}\leq\sum_{\gamma\in\Gamma_{\phi}}\chi(\gamma)k(d_{\mathbb{H}}(z,\gamma
z)).$

The original version of this inequality does not include the character $\chi$
but it does not change the proof.

###### Proof of Proposition 6.1.

Make the assumptions given in the proposition. Let $k_{t}$ be defined by

$k_{t}(\rho)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\mathbf{1}\\{\rho\leq
t\\}}{\sqrt{\cosh t}}.$

Then $k_{t}$ is thought of as the kernel
$k_{t}(x,y)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}k_{t}(d_{\mathbb{H}}(x,y))$
on $\mathbb{H}\times\mathbb{H}$. As such, the convolution
$K_{t}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}k_{t}\ast k_{t}$ makes sense
and corresponds to a compactly supported $K_{t}:[0,\infty)\to\mathbb{R}$. The
Selberg transform is a $*$-morphism, so if $H_{t}$ denotes the Selberg
transform of $K_{t}$, we have

$H_{t}=|h_{t}|^{2}\geq 0$

where $h_{t}$ is the Selberg transform of $k_{t}$. It is proved in [GLMST21,
Proof of Thm. 4.1] that for $t\geq 3$

$h_{t}(r(\lambda_{i}))\gg\sinh\left(t\sqrt{\frac{1}{4}-\lambda}\right).$ (6.1)

We can also estimate

$\displaystyle K_{t}(x,y)$ $\displaystyle=k_{t}\ast k_{t}(x,y)=\frac{1}{\cosh
t}\int_{z\in\mathbb{H}}\mathbf{1}\\{d_{\mathbb{H}}(x,z)\leq
t,d_{\mathbb{H}}(z,y)\leq t\\}$
$\displaystyle\leq\frac{\mathbf{1}\\{d_{\mathbb{H}}(x,y)\leq
2t\\}\mathrm{Vol_{\mathbb{H}}}(B_{\mathbb{H}}(x,t))}{\cosh
t}\ll\mathbf{1}\\{d_{\mathbb{H}}(x,y)\leq 2t\\}.$

We now apply Lemma 6.2 with this pair of functions $K_{t},H_{t}$ and

$t=\frac{1}{2}c\log n$

with $c$ as in GTF($c\log n$). Supposing $f$ is an $L^{2}$-normalized
eigensection of $\mathcal{L}_{\chi}$ with eigenvalue $\lambda$ we obtain for
any $z\in\mathbb{H}$ that

$|f(z)|^{2}\leq
n^{-c\sqrt{\frac{1}{4}-\lambda}}\sum_{\gamma\in\Gamma_{\phi}}\mathbf{1}\\{d(z,\gamma
z)\leq 2t\\}.$ (6.2)

Now suppose $z\in X_{\phi}$ is fixed — the outcome of the following logic will
not depend on $z$.

By GTF($c\log n$) the collection of $\gamma$ for which the right hand side of
(6.2) is not zero lie inside a (possibly trivial) cyclic subgroup of
$\Gamma_{\phi}$. Let $\Gamma^{\prime}$ denote this cyclic subgroup. Suppose it
is not trivial (the trivial case is even easier).

First suppose the non-trivial elements of this $\Gamma^{\prime}$ are
hyperbolic. If $d(z,\gamma z)\leq 2t$ then the displacement length of $\gamma$
is $\leq 2t$. But displacement length is a homomorphism

$\Gamma^{\prime}\to(\mathbb{R},+)$

and the generator of $\Gamma^{\prime}$ has displacement length bounded below
by $\ell_{0}$, the shortest length of a closed geodesic on $X$. This means the
right hand side of (6.2) is at most

$n^{-c\sqrt{\frac{1}{4}-\lambda}}\frac{4t+1}{\ell_{0}}\ll
n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}$

for any $0<c^{\prime}<c$.

Now suppose the non-trivial elements of $\Gamma^{\prime}$ are parabolic, and
let $\gamma_{0}$ be a generator of $\Gamma^{\prime}$. We split into two cases.

Case 1. $z$ is not in $K_{\phi}$.

This means $z$ is in a cusp of $X_{\phi}$ bounded by a horocycle of length $L$
lying over a component of $\partial K$. After changing coordinates to the
standard model of the cusp (cf. $\S$2), $\gamma_{0}(z)=z+L$ where $L$ is the
length of the closed horocycle in $X_{\phi}$ covered by
$\mathrm{Im}(z)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}1$. In this model,
$z\in X_{\phi}$ corresponds to $z=x+iy(z)$ with $y(z)\geq 1$. In this case,
the previous argument does not work because

$d_{\mathbb{H}}(z,\gamma_{0}^{n}z)=2\sinh^{-1}\left(\frac{|n|L}{2y(z)}\right)$
(6.3)

and so $4t+1$ above gets replaced by something exponential in $t$. But we note
for later that $d_{\mathbb{H}}(z,\gamma_{0}^{n}z)\leq 2t$ implies

$n\leq\frac{2y(z)e^{2t}}{L}.$ (6.4)

Instead, we depart from [GLMST21] and re-estimate $K_{t}$. This is already
done in Gamburd [Gam02, Prop. 5.1] and gives

$K_{t}(z,w)\ll\frac{e^{t}}{\cosh
t}\mathbf{1}\\{d_{\mathbb{\mathbb{H}}}(z,w)\leq
2t\\}e^{-\frac{1}{2}d_{\mathbb{H}}(z,w)}\ll\mathbf{1}\\{d_{\mathbb{\mathbb{H}}}(z,w)\leq
2t\\}e^{-\frac{1}{2}d_{\mathbb{H}}(z,w)}.$

Now $\gamma_{0}^{n}$ with $n\neq 0$ contributes to the right hand side of the
pre-trace inequality at most (using (6.4) and (6.3))

$\mathbf{\ll
1}\left\\{|n|\leq\frac{2y(z)e^{2t}}{L}\right\\}\frac{y(z)}{|n|L}\ll\frac{y(z)}{L}\mathbf{1}\left\\{|n|\leq\frac{2y(z)e^{2t}}{L}\right\\}\frac{1}{|n|}.$

Summing over all $n$ (counting one for the identity) gives at most

$\ll 1+\frac{y(z)}{L}\log\left(\frac{2y(z)e^{2t}}{L}\right)$

for the right hand side of pre-trace inequality. Putting things together with
$t=\frac{c}{2}\log n$ gives

$|f(z)|^{2}\leq
n^{-c\sqrt{\frac{1}{4}-\lambda}}\left(1+\frac{y(z)}{L}\log\left(\frac{2y(z)n^{c}}{L}\right)\right).$

Case 2. $z\in K_{\phi}.$

Take all lifts of $\partial K$ to $\mathbb{H}$, where they form a countably
infinite collection of disjoint horocircles. Each such horocircle is tangent
to $\partial\mathbb{H}$ at a fixed point of a parabolic element of $\Gamma$,
and this gives a one-to-one correspondence between infinite cyclic parabolic
subgroups and horocircles. Our assumption on $z$ implies $z$ lifts to a point
disjoint from all closed discs bounded by these horocircles.

In this case we can still change coordinates in the universal cover so
$\gamma_{0}(z)=z+L$, $L\in\pm\mathbb{N}$, and $z=x+iy$ with $y\leq 1$. Then
running the same argument as in the previous case, we get

$\displaystyle|f(z)|^{2}$ $\displaystyle\leq
n^{-c\sqrt{\frac{1}{4}-\lambda}}\left(1+\frac{y(z)}{L}\log\left(\frac{2y(z)n^{c}}{L}\right)\right)\ll
n^{-c\sqrt{\frac{1}{4}-\lambda}}\log\left(n^{c}\right)$ $\displaystyle\ll
n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$

∎

##  7 Bumping off the cusps

We need the following localization formula due to Ismagilov–Morgan–Simon–Sigal
[CFKS87, Thm. 3.2].

######  Theorem 7.1 (IMS localization formula).

Suppose $(M,g)$ is a Riemannian manifold and $\\{J_{i}\\}_{i\in\mathcal{I}}$
are a family of smooth functions $J_{i}:M\to[0,1]$ such that

  1. 1.

$\sum_{i\in\mathcal{I}}J_{i}^{2}\equiv 1$

  2. 2.

on any compact set $K\subset M$, only finitely many $J_{i}$ are not zero

  3. 3.

$\sup_{x\in M}\sum_{i\in\mathcal{I}}|\nabla J_{i}(x)|^{2}<\infty$.

Then

$\Delta=\sum_{i\in\mathcal{I}}J_{i}\Delta J_{i}-\sum_{i\in\mathcal{I}}|\nabla
J_{i}|^{2}.$

Above, $\Delta$ and $\nabla$ are the (non-negative) Laplace-Beltrami and
gradient operators defined w.r.t. $g$.

Suppose that we are given $X_{\phi}$ satisfying Connected and GTF($c\log n$).
Given $\chi\in\mathrm{Hom}(\Gamma_{\phi},\mu_{2})$, let $f$ be an
$L^{2}$-normalized section of $\mathcal{L}_{\chi}$ on $X_{\phi}$ with
eigenvalue $\lambda$ in $[0,\lambda_{0}]$ with $\lambda_{0}<\frac{1}{4}$. Let
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be connected components of
$X_{\phi}\backslash K_{\phi}$, i.e. cusps. Let $L_{1}$ and $L_{2}$ denote the
lengths of the closed horocycles bounding the respective $\mathcal{C}_{i}$.
Fix universal $J:\mathbb{R}\to[0,1]$ smooth and such that

$\displaystyle J(r)$ $\displaystyle\equiv 1\quad r\geq 3,$ (7.1)
$\displaystyle J(r)$ $\displaystyle\equiv 0\quad r\leq 2,$ (7.2)
$\displaystyle\sqrt{1-J^{2}}$ $\displaystyle\in C^{\infty}.$ (7.3)

In the standard model of $\mathcal{C}_{i}$ (cf. $\S$2), let

$J_{i}(z)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}J\left(\frac{y(z)}{L_{i}}\right)$

define a smooth cutoff on $\mathcal{C}_{i}$ — then $J_{1}$ and $J_{2}$ have
disjoint supports. Extend both functions by zero to the whole of $X_{\phi}$.
Let

$J_{0}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sqrt{1-\left(J_{1}^{2}+J_{2}^{2}\right)}$

and

$f^{\prime}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}J_{0}f.$

We aim to prove that $f^{\prime}$ has close to the same Rayleigh quotient as
$f$. That is, we compare

$\frac{\int_{X_{\phi}}|\nabla
f|^{2}}{\int_{X_{\phi}}|f|^{2}},\,\,\,\frac{\int_{X_{\phi}}|\nabla
f^{\prime}|^{2}}{\int_{X_{\phi}}|f^{\prime}|^{2}}.$

The estimates involving $|f|^{2}$ and $|f^{\prime}|^{2}$ integrals will rely
on the following result of Gamburd [Gam02, Lemma 4.1].

######  Theorem 7.2 (Gamburd’s Collar Lemma).

Suppose $\lambda_{0}\in(0,\frac{1}{4})$. There exists $c_{\lambda_{0}}>0$ such
that for any $\phi,\chi,f$ as above, using coordinate $y(z)$ in a fixed cusp
$\mathcal{C}_{i}$ of $X_{\phi}$

$\int_{y(z)=L_{i}}^{2L_{i}}|f|^{2}\geq
c_{\lambda_{0}}\int_{y(z)=2L_{i}}^{\infty}|f|^{2}$

(To obtain this statement from Gamburd’s result without thinking about line
bundles, one can apply [Gam02, Lemma 4.1] to $f$ viewed as a function on
$X_{\phi,\chi}$, and work in the double cover of the cusp.)

It follows from Theorem 7.2 since in $\mathcal{C}_{i}$, $J_{0}\equiv 1$ in
$y(z)\leq 2L_{i}$ that

$\|f^{\prime}\|^{2}\geq\int_{y(z)=L_{i}}^{2L_{i}}|f|^{2}\geq
c_{\lambda_{0}}\int_{y(z)=2L_{i}}^{\infty}|f|^{2}\geq
c_{\lambda_{0}}\|J_{i}f\|^{2}.$

Adding this for $i=1,2$, using
$\|J_{1}f\|^{2}+\|J_{2}f\|^{2}=1-\|f^{\prime}\|^{2}$ and rearranging gives

$\|f^{\prime}\|^{2}\geq\frac{c_{\lambda_{0}}}{2+c_{\lambda_{0}}}>0.$ (7.4)

We now start the main argument of this $\S$.

######  Proposition 7.3.

With notation as above, if GTF($c\log n$) holds, then for any $c^{\prime}<c$

$\frac{\langle\Delta
f^{\prime},f^{\prime}\rangle}{\|f^{\prime}\|^{2}}-\lambda\ll_{c^{\prime},\lambda_{0}}n^{-2c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$

###### Proof.

Let

$\mathfrak{J}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}|\nabla
J_{0}|^{2}+|\nabla J_{1}|^{2}+|\nabla J_{2}|^{2}\geq 0.$

This is supported in $X_{\phi}\backslash K_{\phi}$ and in $\mathcal{C}_{i}$,
$\mathfrak{J}$ is supported on $\\{z:2L_{i}\leq y(z)\leq 3L_{i}\\}$. By
Theorem 7.1 we have

$\displaystyle\langle\Delta f^{\prime},f^{\prime}\rangle$
$\displaystyle=\langle\mathfrak{J}f,f\rangle+\langle\Delta
f,f\rangle-\langle\Delta J_{1}f,J_{1}f\rangle-\langle\Delta
J_{2}f,J_{2}f\rangle$
$\displaystyle=\langle\mathfrak{J}f,f\rangle+\lambda-\langle\Delta
J_{1}f,J_{1}f\rangle-\langle\Delta J_{2}f,J_{2}f\rangle.$

By lifting, each $J_{i}f$, $i=1,2$, can be viewed as a smooth compactly
supported function on a parabolic cylinder (extending a cusp region of
$X_{\tilde{\phi}})$. As the spectrum of the $L^{2}$ Laplacian in this cylinder
is bounded below by $\frac{1}{4}$ it follows as in [HM23, Lemma 4.2] that

$\langle\Delta J_{i}f,J_{i}f\rangle\geq\frac{1}{4}\|J_{i}f\|_{2}^{2}\quad
i=1,2.$

Inserting this above and rearranging gives

$\displaystyle 0\leq\langle\Delta f^{\prime},f^{\prime}\rangle$
$\displaystyle\leq\langle\mathfrak{J}f,f\rangle+\lambda-\frac{1}{4}\left(\|J_{1}f\|_{2}^{2}+\|J_{2}f\|_{2}^{2}\right)$
$\displaystyle=\langle\mathfrak{J}f,f\rangle+\lambda\|J_{0}f\|_{2}^{2}+\left(\lambda-\frac{1}{4}\right)\left(\|J_{1}f\|_{2}^{2}+\|J_{2}f\|_{2}^{2}\right)$
(7.5)
$\displaystyle\leq\langle\mathfrak{J}f,f\rangle+\lambda\|f^{\prime}\|_{2}^{2}.$
(7.6)

Now we estimate

$|\langle\mathfrak{J}f,f\rangle|\leq\|\mathfrak{J}\|_{1}\max\left(\sup_{z\in
C_{1},y(z)\leq 3L_{1}}|f(z)|^{2},\sup_{z\in C_{2},y(z)\leq
3L_{2}}|f(z)|^{2}\right)$

By Proposition 6.1 we have

$\sup_{z\in C_{i},y(z)\leq
3L_{i}}|f(z)|^{2}\ll_{c}n^{-c\sqrt{\frac{1}{4}-\lambda}}\left(1+\log\left(6n^{c}\right)\right)\ll_{c^{\prime}}n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}$

for any $c^{\prime}<c$. Hence

$|\langle\mathfrak{J}f,f\rangle|\ll_{c^{\prime}}\|\mathfrak{J}\|_{1}n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$
(7.7)

We hence turn to $\|\mathfrak{J}\|_{1}$. In $\mathcal{C}_{1}$

$\displaystyle\mathfrak{J}$ $\displaystyle=|\nabla J_{0}|^{2}+|\nabla
J_{1}|^{2}=y^{2}\left(\frac{\partial J_{0}}{\partial
y}\right)^{2}+y^{2}\left(\frac{\partial J_{1}}{\partial y}\right)^{2}.$

Since derivatives of $J$ and $\sqrt{1-J^{2}}$ are uniformly bounded

$\left(\frac{\partial J_{0}}{\partial y}\right)^{2},\left(\frac{\partial
J_{1}}{\partial y}\right)^{2}\ll\frac{1}{L_{1}^{2}}.$

Hence

$\mathfrak{J}\ll\frac{y^{2}}{L_{1}^{2}}.$

Since $\mathfrak{J}$ is supported on $y$ between $2L_{1}$ and $3L_{1}$ we have

$\|\mathfrak{J}\|_{L^{1}(\mathcal{C}_{1})}\ll\int_{0}^{L_{1}}\int_{2L_{1}}^{3L_{1}}\frac{y^{2}}{L_{1}^{2}}\frac{dy\wedge
dx}{y^{2}}=1.$

Exactly the same bound holds in $\mathcal{C}_{2}$ and hence for
$\|\mathfrak{J}\|_{1}$, so

$|\langle\mathfrak{J}f,f\rangle|\ll_{c^{\prime}}n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$

Using this in (7.6) gives

$\langle\Delta
f^{\prime},f^{\prime}\rangle-\lambda\|f^{\prime}\|^{2}\ll_{c^{\prime}}n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}};$

now dividing by $\|f^{\prime}\|^{2}$ and using (7.4) gives

$\frac{\langle\Delta
f^{\prime},f^{\prime}\rangle}{\|f^{\prime}\|^{2}}-\lambda\ll_{c^{\prime},\lambda_{0}}n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$
(7.8)

∎

##  8 Bumping to zero along tile edge

Recall that $\mathcal{K}$ is the fixed compact part of the fundamental domain
for $\Gamma$. Let $\mathcal{G}$ denote the set of lengths of geodesic boundary
segments of $\mathcal{K}$. Now let $\mathcal{E}$ denote a infinite geodesic
edge of the tiling of $X_{\phi}$ by $F$-tiles. This edge meets one or two
cusps. The case of one cusp is similar so suppose there are two, and call them
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ as in the previous section bounded by
closed horocycles of lengths $L_{1}$ and $L_{2}$ respectively.

Let
$\kappa\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{2}\sinh^{-1}\left(\frac{1}{8}\right)$
and
$\Omega\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{2}\ell(\mathcal{E}\cap
K_{\phi})\in\frac{1}{2}\mathcal{G}$. We will use Fermi coordinates in a
neighborhood of $\mathcal{E}$ of the form

$(\rho,t)\in[-\kappa,\kappa]\times\left[-(\Omega+\log(4L_{2})),(\Omega+\log(4L_{1}))\right]\xrightarrow{\theta}X_{\phi}$

where $\rho$ is the signed distance to $\mathcal{E}$ and $t$ is the (signed)
arc-length along $\mathcal{E}$: here we pick a point $p$ on $\mathcal{E}$
midway between the two boundary components of $K_{\phi}$ that $\mathcal{E}$
intersects for which $t(p)=0$ and we have oriented $\mathcal{E}$ from
$\mathcal{C}_{2}$ to $\mathcal{C}_{1}$ in some way to fix the sign of $\rho$.

Let $\mathcal{T}$ denote the image of this map $\theta$; the construction of
the constants we use ensure that $\theta$ is a diffeomorphism to its image. In
these coordinates the hyperbolic metric is

$ds^{2}=d\rho^{2}+(\cosh\rho)^{2}dt^{2}.$

Figure 8.1: Illustration of Fermi tube $\mathcal{T}$

Given $f$ as in the previous section, let $f^{\prime}=J_{0}f$ be the function
obtained from the previous section. Let $K_{\phi}^{+}$ denote the union of
$K_{\phi}$ and the regions of $\mathcal{C}_{i}$ with $y(z)\leq 4L_{i}$ in the
standard model of each cusp. Then

$K_{\phi}^{+}\supseteq\mathcal{T}.$

Let $J$ be as before, satisfying (7.1)–(7.3). In the above Fermi coordinates
in $\mathcal{T}$, let
$J^{*}(\rho,t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}J\left(\frac{2|\rho|}{\kappa}\right)$for
$|\rho|\leq\kappa$ and extend by constant value one to a function on
$K_{\phi}^{+}$. Both $J^{*}$ and $\sqrt{1-(J^{*})^{2}}$ are smooth. Now,
$J_{0}J^{*}$ is smooth on $K_{\phi}^{+}$ and extends (by zero) to a smooth
function on all of $X_{\phi}$.

Let $f^{\prime\prime}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}J_{0}J^{*}f$
and we will write $J^{*}f^{\prime}$ for this, understanding what it means
(understanding the product only requires understanding $J^{*}$ in the support
of $J_{0}$).

######  Proposition 8.1.

With notation as above, for any $c^{\prime}<c$ and $\lambda<\lambda_{0}$

$\frac{\langle\Delta f^{\prime\prime},f^{\prime\prime}\rangle}{\langle
f^{\prime\prime},f^{\prime\prime}\rangle}-\lambda\ll_{c^{\prime},\lambda_{0}}n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$

###### Proof.

Let

$Q\stackrel{{\scriptstyle\mathrm{def}}}{{=}}J_{0}^{2}\left(|\nabla
J^{*}|^{2}+|\nabla\sqrt{1-J^{*2}}|^{2}\right)\geq 0,$

$Q$ is supported in $\mathcal{T}$. Moreover, $Q$ is smooth and bounded in
$\mathcal{T}$ (independently of $\mathcal{E}$).

The IMS localization formula (Theorem 7.1) applied to $f^{\prime}$ tells us
that

$\displaystyle\langle\Delta f^{\prime\prime},f^{\prime\prime}\rangle$
$\displaystyle=\langle\Delta f^{\prime},f^{\prime}\rangle+\langle
Qf,f\rangle-\langle\Delta\sqrt{1-J^{*2}}f^{\prime},\sqrt{1-J^{*2}}f^{\prime}\rangle$
$\displaystyle\leq\langle\Delta f^{\prime},f^{\prime}\rangle+\langle
Qf,f\rangle.$

Since $Q$ is supported in $\mathcal{T}$,

$\langle
Qf,f\rangle\ll\mathrm{area}(\mathcal{T})\|f\lvert_{\mathcal{T}}\|_{\infty}^{2}.$
(8.1)

The area of $\mathcal{T}$ is

$\displaystyle\int_{-\kappa}^{\kappa}\int_{-\Omega-\log(4L_{2})}^{\Omega+\log(4L_{1})}\cosh\rho
dtd\rho$
$\displaystyle=\left(2\Omega+\log(4L_{1})+\log(4L_{2})\right)\int_{-\kappa}^{\kappa}\cosh\rho
d\rho$ $\displaystyle\ll\left(2\Omega+\log(4L_{1})+\log(4L_{2})\right)$
$\displaystyle\ll\log n$ (8.2)

where the last inequality used both $L_{1},L_{2}\leq n$.

Using Proposition 6.1 and (8.2) in (8.1) gives for $c^{\prime}<c$

$\langle Qf,f\rangle\ll_{c^{\prime\prime},\lambda_{0}}(\log
n)n^{-c^{\prime\prime}\sqrt{\frac{1}{4}-\lambda}}\ll_{c^{\prime},\lambda_{0}}n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$

Hence

$\langle\Delta f^{\prime\prime},f^{\prime\prime}\rangle-\langle\Delta
f^{\prime},f^{\prime}\rangle\ll_{c^{\prime},\lambda_{0}}n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$
(8.3)

We also have

$\langle f^{\prime\prime},f^{\prime\prime}\rangle=\langle
f^{\prime},f^{\prime}\rangle-\langle
J_{0}\sqrt{1-J^{*2}}f,J_{0}\sqrt{1-J^{*2}}f\rangle$

and since $J_{0}\sqrt{1-J^{*2}}$ is supported on $\mathcal{T}$ and universally
bounded, repeating the same arguments as before gives

$\langle f^{\prime\prime},f^{\prime\prime}\rangle-\langle
f^{\prime},f^{\prime}\rangle\gg_{c^{\prime},\lambda_{0}}-n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}.$
(8.4)

Combining the above gives

$\displaystyle\frac{\langle\Delta
f^{\prime\prime},f^{\prime\prime}\rangle}{\langle
f^{\prime\prime},f^{\prime\prime}\rangle}-\frac{\langle\Delta
f^{\prime},f^{\prime}\rangle}{\langle f^{\prime},f^{\prime}\rangle}=$
$\displaystyle\frac{1}{\langle
f^{\prime},f^{\prime}\rangle}\left(\langle\Delta
f^{\prime\prime},f^{\prime\prime}\rangle-\langle\Delta
f^{\prime},f^{\prime}\rangle\right)$ $\displaystyle+\frac{\langle\Delta
f^{\prime\prime},f^{\prime\prime}\rangle}{\langle
f^{\prime\prime},f^{\prime\prime}\rangle\langle
f^{\prime},f^{\prime}\rangle}\left(\langle
f^{\prime},f^{\prime}\rangle-\langle
f^{\prime\prime},f^{\prime\prime}\rangle\right)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:mass-
estimate}}}{{\ll_{c^{\prime},\lambda_{0}}}}$ $\displaystyle\max(\langle\Delta
f^{\prime\prime},f^{\prime\prime}\rangle-\langle\Delta
f^{\prime},f^{\prime}\rangle,0)$ $\displaystyle+\frac{\langle\Delta
f^{\prime\prime},f^{\prime\prime}\rangle}{\langle
f^{\prime\prime},f^{\prime\prime}\rangle}\max(\langle
f^{\prime},f^{\prime}\rangle-\langle
f^{\prime\prime},f^{\prime\prime}\rangle,0)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:energy-second},\eqref{eq:mass-
second}}}{{\ll_{c^{\prime},\lambda_{0}}}}$ $\displaystyle
n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}\left(1+\frac{\langle\Delta
f^{\prime\prime},f^{\prime\prime}\rangle}{\langle
f^{\prime\prime},f^{\prime\prime}\rangle}\right).$

This implies now rearranging and using Proposition 7.3 that

$\displaystyle\frac{\langle\Delta
f^{\prime\prime},f^{\prime\prime}\rangle}{\langle
f^{\prime\prime},f^{\prime\prime}\rangle}\leq$
$\displaystyle\left(1+O_{c^{\prime},\lambda_{0}}\left(n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}\right)\right)\frac{\langle\Delta
f^{\prime},f^{\prime}\rangle}{\langle f^{\prime},f^{\prime}\rangle}$
$\displaystyle+O_{c^{\prime},\lambda_{0}}\left(n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}\right)$
$\displaystyle\leq$
$\displaystyle\left(1+O_{c^{\prime},\lambda_{0}}\left(n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}\right)\right)\left(\lambda+O_{c^{\prime},\lambda_{0}}\left(n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}\right)\right)$
$\displaystyle+O_{c^{\prime},\lambda_{0}}\left(n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}\right)$
$\displaystyle=\lambda+O_{c^{\prime},\lambda_{0}}\left(n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}\right).$

∎

##  9 Proof of Theorems 1.2 and 1.1

###### Proof of Theorem 1.2.

Suppose that $\phi$ satisfies Connected and
GTF($c\log n$) for $c>0$. Results of previous sections establish these hold
a.a.s.

Let $G_{\phi}$ denote the $4$-regular Schreier graph constructed from the side
pairing congruences $A,B$ of the fundamental domain $F$ acting on $[n]$ via
$\phi$. Since we assume $X_{\phi}$ is connected, it is realized as
$\Gamma_{\phi}\backslash\mathbb{H}$ and moreover $G_{\phi}$ is connected. This
graph $G_{\phi}$ encodes the dual structure of the tiling of $X_{\phi}$ by
$\mathcal{K}$-tiles. It has a distinguished vertex $\otimes$ corresponding to
the tile labeled ‘1’ under
$X_{\phi}=\Gamma\backslash_{\phi}\left(\mathbb{H}\times[n]\right)$. Let
$o_{n}$ denote an element of this tile. Since $X_{\phi}$ deformation retracts
to a embedded copy of $G_{\phi}$, we have (via this deformation retract)
$\pi_{1}(G_{\phi},\otimes)\cong\pi_{1}(X_{\phi},o_{n})$=$\Gamma_{\phi}$.

The edges of $G_{\phi}$ are directed and labeled by $A$ and $B$. Let $T$
denote a spanning tree for $G_{\phi}$. Gluing $\mathcal{K}$-tiles according to
$T$ gives a fundamental domain $F_{\phi}$ for $\Gamma_{\phi}$ in $\mathbb{H}$.
Arbitrarily direct the elements of $E(G_{\phi}\backslash T)$ (one could take
the previously given directions, for example). For each $e\in
E(G_{\phi}\backslash T)$ there is a unique homotopy class of loop in
$G_{\phi}$ beginning and ending at $\otimes$ that proceeds (uniquely) to the
start point of $e$ in $T$, traverses $e$ in its given direction, then returns
(uniquely) to $\otimes$ in $T$. These loops gives a free basis
$\mathcal{B}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{\gamma_{e}:e\in
E(G_{\phi}\backslash T)\\}$ of the free group
$\Gamma_{\phi}\cong\pi_{1}(G_{\phi},\otimes)$. This is the basis described in
Theorem 1.2. These elements are the side pairing congruences for $F_{\phi}$.

Let $\chi_{1}$ and $\chi_{2}$ be as in the statement of Theorem 1.2 with
Hamming distance one w.r.t. $\mathcal{B}$. There is a one-to-one differential-
operator-respecting correspondence between smooth sections of
$\mathcal{L}_{\chi_{i}}$ on $X_{\phi}$ and smooth functions $f$ on
$\mathbb{H}$ such that

$f\circ\gamma=\chi_{i}(\gamma)f;\quad\gamma\in\Gamma_{\phi}$ (9.1)

as discussed in $\S$2. This correspondence implies also that for any
differential operator $D$ on $\mathbb{H}$ that

$D[f\circ\gamma_{e}]=\chi_{i}(\gamma_{e})D[f];\quad e\in E(G_{\phi}\backslash
T).$ (9.2)

These functions are uniquely determined by their restriction to
$\overline{F_{\phi}}$ where they satisfy (9.2) for $D$ defined in any
neighborhood of $\partial F_{\phi}$ and such that both sides of (9.2) can be
evaluated with arguments in $\partial F_{\phi}$. (This condition is intended
to make sure that the function on $\overline{F_{\phi}}$ extends to a __ smooth
$\Gamma_{\phi}$-periodic function on $\mathbb{H}$.) The conditions (9.2) are
an ensemble of boundary conditions, one for each side-pairing congruence
$\gamma_{e}$ of $F_{\phi}$.

If $\chi_{1}$ and $\chi_{2}$ have Hamming distance one with respect to
$\mathcal{B}$, there is a unique $e_{0}\in E(G_{\phi}\backslash T)$ such that
$\chi_{1}$ and $\chi_{2}$ differ at $\gamma_{e_{0}}$. Let $\mathcal{E}$ denote
the infinite geodesic in $X_{\phi}$ corresponding to the edge $e_{0}$, and let
$f$ be the eigensection for $\mathcal{L}_{\chi_{1}}$ corresponding to the
bottom of its spectrum, $\lambda_{1}(\mathcal{L}_{\chi_{1}})$. Let
$f^{\prime\prime}$ be the section of $\mathcal{L}_{\chi_{1}}$ that is the
output of the previous $\S$7–8 applied to $f$.

The fact that $f^{\prime\prime}$ is zero in a neighborhood of $\mathcal{E}$
means that, using the previously mentioned correspondences, if we restrict
$f^{\prime\prime}$ to $F_{\phi}$ and then lift it to a section of
$\mathcal{L}_{\chi_{2}},$ we obtain a section $F$ of $\mathcal{L}_{\chi_{2}}$
with (by Proposition 8.1)

$\frac{\langle\Delta F,F\rangle}{\langle
F,F\rangle}\leq\lambda_{1}(\mathcal{L}_{\chi_{1}})+O_{c^{\prime},\lambda_{0}}\left(n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}\right).$

By the variational principle this implies

$\lambda_{1}(\mathcal{L}_{\chi_{2}})\leq\lambda_{1}(\mathcal{L}_{\chi_{1}})+O_{c^{\prime},\lambda_{0}}\left(n^{-c^{\prime}\sqrt{\frac{1}{4}-\lambda}}\right).$

∎

###### Proof of Theorem 1.1.

Given any $\eta>0$ — the ‘scale’ at which we want to find many $\lambda$ below
$\frac{1}{4}$ — let $c>0$ and $C\left(\frac{1}{4}-\eta\right)>0$ be as in
Theorem 1.2.

Using the results of $\S$3 we can find $n>0$ such that

$C\left(\frac{1}{4}-\eta\right)n^{-c\sqrt{\eta}}<\eta$

and $\phi\in\mathrm{Hom}(\Gamma,S_{n})$ such that $X_{\phi}$ is connected,
$X_{\phi}=\Gamma_{\phi}\backslash\mathbb{H}$, and find a generating set
$\mathcal{B}$ of $\Gamma_{\phi}$ that satisfies the conclusion of Theorem 1.2
applied with $\lambda_{0}=\frac{1}{4}-\eta$. Moreover, we can ensure that for
some fixed $\theta\in\mathrm{Hom}(\Gamma,\mu_{2})$

$\lambda_{1}(X_{\phi,\chi(\theta)})\geq\frac{1}{4}-\eta$ (9.3)

(since Spectral Gap holds a.a.s.). let $\chi_{0}$ denote the trivial
representation of $\Gamma_{0}$, so sections of $\mathcal{L}_{\chi_{0}}$
correspond to functions on

$X_{\phi,\chi_{0}}\cong X_{\phi}\sqcup X_{\phi}$

that are odd under the deck transformation swapping components. Hence
$\lambda_{1}(\mathcal{L}_{\chi_{0}})=0$ since there is a locally constant
section of $\mathcal{L}_{\chi_{0}}$.

According to the output of Theorem 1.2, when we move from $\chi_{0}$ to
$\chi_{\theta}$ along a geodesic w.r.t. $d_{\mathsf{Ham}}^{\mathcal{B}}$, at
each step, $\lambda_{1}(\mathcal{L}_{\bullet})$ increases by at most $\eta.$
But this value begins at zero and ends up at least $\frac{1}{4}-\eta$, so its
values must be $\eta$-dense in $[0,\frac{1}{4}]$.

Finally, if $\chi_{0},\chi_{1},\ldots,\chi_{r}=\chi(\theta)$ are the result of
moving along this geodesic, then

$\displaystyle\mathrm{spec}(\Delta_{\mathcal{L}_{\chi_{i}}})\subset\mathrm{spec}(\Delta_{X_{\phi,\chi_{i}}})$
$\displaystyle=\mathrm{spec}(\Delta_{X_{\phi}})\cup\mathrm{spec}(\Delta_{\mathcal{L}_{\chi_{i}}})$
$\displaystyle\subseteq\big{[}\frac{1}{4}-\eta,\infty\big{)}\cup\mathrm{spec}(\Delta_{\mathcal{L}_{\chi_{i}}})$

from (9.3), so the corresponding $\lambda_{1}(X_{\phi,\chi_{i}})$ are also
$\eta$-dense in $\left[0,\frac{1}{4}\right]$. ∎

##  Appendix A Common fixed points of subgroups under random homomorphisms to
permutation groups

Let $\Gamma$ denote a finitely generated free group with a fixed basis
$\mathcal{B}$ and $H$ a finitely generated (necessarily free) subgroup. For
$\phi\in\mathrm{Hom}(\Gamma,S_{n})$, let $\mathrm{fix}_{H}$ denote the number
of points in $[n]$ fixed by every element of $\phi(H)\leq S_{n}$. For a
function $F$ on $\mathrm{Hom}(\Gamma,S_{n})$ we write $\mathbb{E}_{n}[F]$ for
the expected value of $F$ with respect to the uniform probability measure on
the finite set $\mathrm{Hom}(\Gamma,S_{n})$. Let $\ell_{\mathcal{B}}(H)$
denote the minimum, over all bases $\mathcal{B}^{\prime}$ of $H$, of the sum
of word lengths of $\mathcal{B}^{\prime}$ w.r.t. $\mathcal{B}$.

######  Proposition A.1.

If $\mathrm{rank}(H)\geq 2$ and $n\geq\ell_{\mathcal{B}}(H)^{3}$, then

$\mathbb{E}_{n}(\mathrm{fix}_{H})\ll\frac{\ell_{\mathcal{B}}(H)^{6}}{n}.$

###### Proof.

Let $m=\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\ell_{\mathcal{B}}(H)$. In
[PP15, Lemma 6.4], Puder and Parzanchevski show that there is a finite set
$\mathcal{Q}_{H}$ of rational functions, depending on $H$, such that:

  * •

$\mathbb{E}_{n}[\mathrm{fix}_{H}]=\sum_{q\in\mathcal{Q}_{H}}q(n)$ for
$n\geq\ell_{\mathcal{B}}(H)$

  * •

each $q\in\mathcal{Q}_{H}$ is of the form

$q(n)=\frac{(n)_{V(q)}}{\prod_{b\in\mathcal{B}}(n)_{E_{b}(q)}}$

where $V(q),E_{b}(q)\leq\ell_{B}(H)$ and $(n)_{a}$ is the falling Pochhammer
symbol. Hence since (e.g. [MNP22, Lemma 5.23]) if $a\leq\frac{1}{2}n$

$n^{a}\left(1-\frac{a^{2}}{n}\right)\leq(n)_{a}\leq n^{a}$ (A.1)

we get for $n\geq 2\ell_{\mathcal{B}}(H)^{2}$ that

$q(n)\asymp n^{V(q)-\sum_{b\in\mathcal{B}}E_{b}(q)}$

in the regime of $n\to\infty$. Let
$\chi(q)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}V(q)-\sum_{b\in\mathcal{B}}E_{b}(q)$.

  * •

For $q\in\mathcal{Q}_{H},$ $\chi(q)\leq 1-\mathrm{rank}(H)$.

In fact, Puder and Parzanchevski give a much more detailed description of the
set $\mathcal{Q}_{H}$ above, but this will not be needed here. It follows
immediately that for fixed $H$,
$\mathbb{E}_{n}[\mathrm{fix}_{H}]=O(n^{\chi(H)})$ but we need a version of
this statement that is uniform over (mildly) varying $H$. It is possible to do
this using the detailed description of $H$ as was done in Puder [Pud15, §5]
but we do not need such fine estimates so can bypass this by analytic methods.
A variant form of the following trick appeared in [MNP22] for a similar
purpose.

Suppose $n\geq m^{3}$ and note from (A.1) that

$m^{3}\geq\mathbb{E}_{m^{3}}[\mathrm{fix}_{H}]\gg\sum_{q\in\mathcal{Q}_{H}}(m^{3})^{\chi(q)}.$

Then supposing $\mathrm{rank}(H)\geq 2$

$\mathbb{E}_{n}[\mathrm{fix}_{H}]\ll\sum_{q\in\mathcal{Q}_{H}}n^{\chi(q)}\leq\sum_{\chi=-1}^{-\infty}m^{3(1-\chi)}n^{\chi}\ll\frac{m^{6}}{n}$

as required. ∎

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Michael Magee,
Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1
3LE Durham, UK
IAS Princeton, School of Mathematics, 1 Einstein Drive, Princeton 08540, USA
<EMAIL_ADDRESS>