# Isospectrality for infinite-type hyperbolic surfaces with discrete length spectrum

Federica Fanoni, David Fisac

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Federica Fanoni, David Fisac. Isospectrality for infinite-type hyperbolic surfaces with discrete length spectrum. 2026. hal-05524785

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# ISOSPECTRALITY FOR INFINITE-TYPE HYPERBOLIC SURFACES WITH DISCRETE LENGTH SPECTRUM

FEDERICA FANONI AND DAVID FISAC

**ABSTRACT.** We prove that every family of isospectral surfaces with discrete length spectrum arising from Sunada's method is finite. Furthermore, by introducing the topological notion of surfaces with self-duplicating ends, we show that every finite group can be realized as the full isometry group of a hyperbolic structure with discrete spectrum on such a surface, if the genus is infinite. Under the same topological assumptions, we also demonstrate that the above-mentioned isospectral families can have unbounded cardinality within a fixed moduli space.

## 1. INTRODUCTION

Given a hyperbolic surface, a classical invariant is its *length spectrum*, that is, the multi-set of lengths of closed geodesics, counted with multiplicities. A natural question is which aspects of the hyperbolic structure are detected by the length spectrum. Said differently, how similar are *isospectral* hyperbolic surfaces (hyperbolic surfaces with the same length spectrum)?

In the closed case, it was conjectured by Gel'fand [Gel63] that the length spectrum completely determines the hyperbolic structure. The conjecture was first disproved by Vignéras [Vig80], and since then lots of work has been put into constructing isospectral families (see Gordon's survey [Gor00] for more details and references). Nonetheless, the length spectrum does determine the topology, as shown by Huber [Hub59], and moreover Wolpert [Wol79] proved that a *generic* surface is determined by its spectrum. Furthermore, there is a bound on the size of an isospectral family which only depends on the topology of the surface — see the work of Buser [Bus10, Chapter 13] and Parlier [Par18].

The case of surfaces of infinite type is strikingly different. In particular, it was shown by the first author in [Fan23] that, for infinite-type surfaces without planar ends, there is no upper bound to the size of isospectral families. If the surface satisfies an additional topological condition (its space of ends is self-similar — see Section 2), there are isospectral families which are (uncountably) infinite.

Another aspect which clearly sets length spectra of finite- and infinite-type surfaces apart is discreteness. Here we say that a surface has *discrete* length spectrum if, for any $L > 0$, there are finitely many closed geodesics of length at most $L$. While the length spectrum of a finite-type hyperbolic surface is always discrete, in the infinite-type case it is very easy to construct structures with non-discrete length spectrum — it is for instance enough to give the same length to all curves in a pants decomposition — while it is harder to construct structures with discrete length spectrum. This is done by Basmajian and Kim in [BK08], where they show that on any infinite-type surface there are infinite-dimensional parameter spaces of hyperbolic structures with discrete (respectively, non-discrete) length spectrum.

The constructions of isospectral families in [Fan23] all yield hyperbolic surfaces with non-discrete length spectrum, and the goal of this paper is to understand what happens if we require the spectrum to be discrete.

<footnote>Date: February 24, 2026.</footnote>
Before stating our results, recall that almost all known constructions of isospectral families rely on a criterion by Sunada [Sun85]:

**Theorem 1.1 ([Sun85]).** Let $X$ be a complete hyperbolic surface, and $G$ a group of isometries with finitely many fixed points. For $H_1, H_2$ two almost conjugate subgroups of $G$ acting without fixed points, the quotient surfaces $X/H_1$ and $X/H_2$ are isospectral.

Note that Theorem 1.1 was originally stated for finite groups $G$ and compact manifolds, but both hypotheses can be dropped — see [Bus10, Chapter 10] and the transplantation of geodesics technique from Buser [Bus86] (see also [Bér92]).

The infinite isospectral families in [Fan23] are constructed using Theorem 1.1. We ask if infinite isospectral families can be constructed in the same way in the case of surfaces with discrete length spectrum. Our first result is a negative answer to this question.

**Theorem A.** Let $X$ be a hyperbolic surface and $G \leq \text{Isom}^+(X)$ acting on it. Let $\{H_i\}_{i \in I}$ be a family of almost conjugate subgroups of $G$ without fixed points. Define a family of hyperbolic surfaces $\{Y_i = X/H_i\}_{i \in I}$. If the $\mathcal{L}(Y_i)$ are discrete, then $I$ is finite.

On the other hand, we can show (using Sunada's criterion) that one can construct arbitrarily large isospectral families, also if we assume the length spectrum to be discrete:

**Theorem B.** Let $S$ be the blooming Cantor tree surface. For every $n \in \mathbb{N}$, there is a hyperbolic structure $X$ on $S$ with discrete length spectrum and such that $|\mathcal{I}(X)| \geq n$.

Here $\mathcal{I}(X)$ is the collection of all hyperbolic structures on $S$ which are isospectral to $X$. The blooming Cantor tree is the surface whose endspace is a Cantor set, and which has no planar ends (see Figure 2). We can actually prove the theorem for a larger class of surfaces: all infinite-genus surfaces $S$ such that $(\text{Ends}(S), \text{Ends}_g(S))$ is homeomorphic to two copies of itself (see Theorem 5.1). We say that such surfaces have self-duplicating endspace.

As in the proof of Theorem B we use Sunada's criterion, Theorems A and B give a pretty complete picture of what can be done with Sunada's construction, under the discreteness assumption and for infinite-genus surfaces with self-duplicating endspace.

To prove Theorem B, we first show that any infinite-genus surface with self-duplicating endspace admits hyperbolic structures with any given finite isometry group:

**Theorem C.** Let $S$ be an infinite-genus surface with self-duplicating endspace and $G$ any finite group. Then there is a hyperbolic structure $X$ on $S$ with discrete length spectrum and $\text{Isom}(X) \simeq G$.

This result completely characterizes the possible isometry groups of hyperbolic structures with discrete length spectrum on infinite-genus surfaces with self-duplicating endspace, because — as shown by Basmajian and Kim [BK08] — the isometry group of a hyperbolic surface with discrete length spectrum is always finite.

It is also interesting to compare Theorem C with the following result by Aougab, Patel and Vlamis [APV21]:

**Theorem 1.2 ([APV21]).** Let $S$ be an infinite-genus surface without planar ends and $G$ a group.

(1) If the endspace of $S$ is self-similar, there is a hyperbolic structure on $S$ whose isometry group is $G$ if and only if $G$ is countable.

(2) If the endspace of $S$ is doubly pointed, then the isometry group of any hyperbolic structure on $S$ is virtually cyclic.

(3) If $S$ contains a compact non-displaceable subsurface, there is a hyperbolic structure on $S$ with isometry group $G$ if and only if $G$ is finite.
We refer to Section 2 for the definitions of self-similar or doubly pointed endspace and of non-displaceable subsurface. Here we only remark that infinite-genus surfaces with self-duplicating endspace can fall in different categories of Theorem 1.2: for instance, the blooming Cantor tree has self-similar endspace, so the discreteness of the length spectrum assumption really changes the isometry groups that can arise. On the other hand, there are infinite-genus surfaces with self-duplicating endspace which admit a compact non-displaceable subsurface: for example, the surface without planar ends whose endspace is $C_1 \cup C_2 \cup P$, where $C_1$ and $C_2$ are Cantor sets and $P$ is a (countable) set of isolated ends whose closure is $P \cup C_2$. For such surfaces, the discreteness assumption doesn't change the class of possible isometry groups. On the other hand, the constructions in [APV21] all yield structures with non-discrete length spectrum, so Theorem C provides a new insight in the problem of understanding isometry groups of infinite-type surfaces also in this case.

We end this introduction with an open question. We have seen that Sunada's method doesn't allow us to construct infinite isospectral families of hyperbolic surfaces with discrete length spectrum. On the other hand, [BK08] showed that there is an abundance of surfaces with discrete length spectrum, so we ask: is there an alternative way to construct isospectral families, which yields infinite ones also in the discrete length spectrum case?

## ACKNOWLEDGEMENTS

The first author acknowledges support of the ANR grants MAGIC (ANR-23-TERC-0007) and GALS (ANR-23-CE40-0001). The second author is funded by the ANR grant GALS (ANR-23-CE40-0001).

## 2. PRELIMINARIES

We introduce here the basic topological, geometric and group-theoretic notions needed in the rest of the paper.

**Topology.** By surface we mean a two-dimensional orientable connected manifold. We also assume that, if the boundary is non-empty, it is compact. As proved by Kerékjártó and Richards in [Ker23, Ric63], a surface $S$ is topologically determined by its genus and number of boundary components — which might be finite or infinite —, and the pair of topological spaces $(\text{Ends}(S), \text{Ends}_g(S))$. Here, $\text{Ends}(S)$ is the space of ends of the surface and $\text{Ends}_g(S)$ is the space of non-planar ends (see Aramayona and Vlamis' survey [AV20] for the definitions).

A surface has *self-similar endspace* (as first defined by Mann and Rafi in [MR19]) if for every decomposition

$$ \text{Ends}(S) = U_1 \sqcup \dots \sqcup U_n $$

into pairwise disjoint clopen subsets $U_i$, there is $i \in \{1, \dots, n\}$ and an open subset $A \subset U_i$ such that

$$ (A, A \cap \text{Ends}_g(S)) \simeq (\text{Ends}(S), \text{Ends}_g(S)). $$

We say that an endspace is *self-duplicating* if $(\text{Ends}(S), \text{Ends}_g(S))$ is homeomorphic to two copies of itself, i.e.

$$ (\text{Ends}(S), \text{Ends}_g(S)) \simeq (\text{Ends}(S) \times \{1, 2\}, \text{Ends}_g(S) \times \{1, 2\}). $$

Note that this can be easily seen to be equivalent to $(\text{Ends}(S), \text{Ends}_g(S))$ being homeomorphic to $n$ copies of itself, for every $n \ge 1$.

An endspace is *doubly pointed* (see [APV21]) if there are exactly two ends with finite homeomorphism group orbit.

Note that:
* if an endspace is self-similar or self-duplicating, it is not doubly pointed;

* endspaces can be both self-similar and self-duplicating (e.g. in the case of a blooming Cantor tree), self-duplicating but not self-similar (e.g. if $\text{Ends}(S) = \text{Ends}_g(S) = C_1 \cup C_2 \cup P$, where $C_1, C_2$ and $P$ are pairwise disjoint, $C_1$ and $C_2$ are Cantor sets and $P$ is a set of isolated points whose closure is $C_2 \cup P$), or self-similar but not self-duplicating (e.g. if $|\text{Ends}(S)| = |\text{Ends}_g(S)| = 1$).

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Blooming Cantor tree surface.</figcaption></figure>

A compact subsurface $K$ of a surface $S$ is non-displaceable if for every $f \in \text{Homeo}(S)$, $f(K) \cap K \neq \emptyset$.

**Geometry.** A hyperbolic structure $X$ on a surface $S$ is a Riemannian metric of constant curvature $-1$. We will assume such a metric to be complete and of the first kind, if $\partial S = \emptyset$, otherwise we ask that each boundary component is totally geodesic and that the metric on the double is complete and of the first kind. We will also call $X$ a hyperbolic surface. Define the Teichmüller space of a surface $S$, denoted Teich($S$), to be space of marked hyperbolic structures on $S$, i.e. the space of hyperbolic structures up to isotopy. The moduli space of $S$, denoted by $\mathcal{M}(S)$, is the space of hyperbolic structures on $S$ up to isometry. Its universal cover is Teichmüller space.

Given a hyperbolic surface $X$, define its length spectrum $\mathcal{L}(X)$ to be the multi-set of lengths of closed geodesics on the surface with multiplicity. We call two hyperbolic surfaces isospectral if their length spectra coincide. Given a hyperbolic surface $X$ with underlying topological surface $S$, we denote by $\mathcal{I}(X)$ the set

$$ \mathcal{I}(X) := \{Y \in \mathcal{M}(S) \mid \mathcal{L}(Y) = \mathcal{L}(X)\} $$

and by $\tilde{\mathcal{I}}(X)$ the lift of $\mathcal{I}(X)$ to Teichmüller space.

In our constructions we will make repeated use of the so-called Collar Lemma (see e.g. [Bus10, Chapter 4]):

**Theorem 2.1 (Collar Lemma).** Let $X$ be a hyperbolic surface and $\alpha$ a simple closed geodesic on it. Then $X$ contains an embedded collar of width $\arcsin h(\frac{1}{\sinh(\ell_{\alpha}(X)/2)})$.

A consequence is that any two simple closed geodesics of length at most $\arcsin h(1)$ are disjoint.
2.1. **Group theory.** The main group-theoretic notion we will use is almost-conjugacy. Given a group $G$ and two subgroups $H_1, H_2 \leq G$, we say that $H_1$ and $H_2$ are almost conjugate if, for every $g \in G$,

$$|C_g \cap H_1| = |C_g \cap H_2|,$$

where $C_g$ denotes the conjugacy class of $g$ in $G$.

It is not hard to show that if $H_1$ and $H_2$ are almost conjugate subgroups of a group $H$, then for any $n \geq 1$ and any $\sigma_1, \sigma_2 : \{1, \dots, n\} \to \{1, 2\}, \prod_{i=1}^n H_{\sigma_1(i)}$ and $\prod_{i=1}^n H_{\sigma_2(i)}$ are almost conjugate subgroups of $H^n$.

We will use a specific triple $H, H_1, H_2$, described in [Bus10, Example 11.2.2]: $H = (\mathbb{Z}/8\mathbb{Z})^* \times \mathbb{Z}/8\mathbb{Z}$, where $(a, b) \cdot (a', b') = (aa', ab' + b)$, and the subgroups are

$$H_1 = \{e = (1, 0), h_1 = (3, 0), h_2 = (5, 0), (7, 0)\}$$

and

$$H_2 = \{(1, 0), h_3 = (3, 4), h_4 = (5, 4), (7, 0)\}.$$

Then $H_1$ and $H_2$ are almost conjugate, but not conjugate.

### 3. FINITENESS OF ISOSPECTRAL FAMILIES GIVEN BY SUNADA'S CONSTRUCTION

In this section we will prove Theorem A, which we recall here:

**Theorem A.** Let $X$ be a hyperbolic surface and $G \leq \text{Isom}^+(X)$ acting on it. Let $\{H_i\}_{i \in I}$ be a family of almost conjugate subgroups of $G$ without fixed points. Define a family of hyperbolic surfaces $\{Y_i = X/H_i\}_{i \in I}$. If $\mathcal{L}(Y_i)$ are discrete, then $I$ is finite.

We will use two lemmas in the proof, the first one being purely group-theoretic.

**Lemma 3.1.** Let $G$ be a group and $\{H_i\}_{i \in I}$ be a family of almost conjugate subgroups of finite index. Then $I$ is finite.

*Proof.* By definition, the subgroups are almost conjugate if, for every conjugacy class $C$ of $G$, then

$$|H_i \cap C| = |H_j \cap C|$$

for all $i, j \in I$. It is a standard fact (see e.g. [Gor00, Proposition 4.1C]) that their permutation representations

$$\rho_{H_i} : G \to \text{Sym}(G/H_i)$$

$$g \mapsto (h \mapsto g \cdot h)$$

are pairwise conjugate for all $i \in I$. Therefore, by elementary computations, their kernels coincide. Denote then

$$K = \ker(\rho_{H_i}) = \bigcap_{g \in G} gH_ig^{-1}$$

for all $i \in I$. Note also that $K$ is a normal subgroup of all $H_i$'s. By considering $H_i/K \leq G/K$, it is straightforward that $[G : K] \leq [G : H_i] < \infty$. Since the subgroups of $G$ containing $K$ are in bijection with the subgroups of $G/K$, which is finite, we conclude. $\square$

**Lemma 3.2.** Let $X$ be a hyperbolic surface and $G < \text{Isom}^+(X)$ acting on it. Then,

$$|\text{Stab}_G(\gamma)| < \infty$$

for all closed geodesics $\gamma$.
*Proof.* Note that $\mathrm{Stab}_G(\gamma) \subseteq \{f \in \mathrm{Isom}(X) | f(\gamma) \cap \gamma \neq \emptyset\}$. Then the result follows from proper discontinuity of the action of $\mathrm{Isom}(X)$ (see e.g. [APV21, Lemma 2.6]). □

*Proof of Theorem A.* Assume first that $\mathcal{L}(X)$ is discrete. By [BK08, Prop 3.4], then $\mathrm{Isom}(X)$ is finite, and hence $[G : H_i]$ are also all finite. In this case the result follows from Lemma 3.1.

Assume now that $\mathcal{L}(X)$ is not discrete. We will now prove that in this case $[G : H_i]$ are still all finite. It is enough to prove it for one of the $H_i$'s, since almost conjugate subgroups have the same index. Take $H$ to be one of them for simplicity.

Fix a closed geodesic $\gamma$ on $X$. Note that there is a natural correspondence between the orbit set of $\gamma$, $G \cdot \gamma$, and the quotient group $G/\mathrm{Stab}_G(\gamma)$. We will define two quotients on the quotient set. First,

$$A_H(\gamma) = G \cdot \gamma / \sim, \text{ where } \alpha \sim \beta \text{ if } h(\alpha) = \beta \text{ for some } h \in H.$$

Note that $A_H(\gamma)$ is finite for all $\gamma$, because of $\mathcal{L}(X)$ being non-discrete and $\mathcal{L}(X/H)$ being discrete. Note now that $A_H(\gamma)$ is also in bijection with the group $G/\approx$, where $g_1 \approx g_2$ if $h(g_1(\gamma)) = g_2(\gamma)$ for some $h \in H$. That is equivalent to $g_1 \in \mathrm{Stab}_G(\gamma)g_2H$. There is a natural bijection between $A_H(\gamma)$ and $\mathrm{Stab}_G(\gamma)\backslash G/H$. Furthermore, by definition,

$$[G : H] = \sum_{\mathrm{Stab}_G(\gamma)gH \in \mathrm{Stab}_G(\gamma)\backslash G/H} [\mathrm{Stab}_G(\gamma) : \mathrm{Stab}_G(\gamma) \cap gHg^{-1}].$$

Note that, since $A_H(\gamma)$ is finite, so is the index set of the sum, and that every term of the sum is finite because of Lemma 3.2. Hence, $[G : H]$ is finite, and therefore we can conclude again by Lemma 3.1. □

## 4. REALIZING FINITE GROUPS AS ISOMETRY GROUPS

The goal of this section is to prove that all finite groups can be realized as isometry groups of hyperbolic structures with discrete length spectrum on a large class of surfaces:

**Theorem C.** Let $S$ be an infinite-genus surface with self-duplicating endspace and $G$ any finite group. Then there is a hyperbolic structure $X$ on $S$ with discrete length spectrum and $\mathrm{Isom}(X) \simeq G$.

**Remark 4.1.** Note that every hyperbolic structure with discrete length spectrum has finite isometry group (see [BK08, Proposition 3.4]), so our result completely classifies the possible isometry groups of hyperbolic structures with discrete length spectrum on an infinite-genus surface with self-duplicating endspace.

We will use the following result, which follows from the work of Basmajian and Kim [BK08]:

**Lemma 4.2.** Let $S$ be a surface with a single boundary component and let $\varepsilon > 0$. Then there is a hyperbolic structure $X$ on $S$ with discrete length spectrum and such that the boundary component has length $\varepsilon$.

*Proof.* It is enough to observe that the construction by Basmajian and Kim [BK08] can be easily adapted to include one geodesic boundary component, and that one can freely choose its length $\varepsilon$. □

**Remark 4.3.** The orthospectrum of any (finite- or infinite-type) hyperbolic surface with compact boundary is discrete. One way of seeing this is to double the surface along the boundary and use the fact that the multiset of length of geodesics intersecting a given compact set (in this case, what used to be the boundary) is discrete.
*Proof of Theorem C.* Fix $S$ as in the statement, a finite group $G$ and $\varepsilon \in (0, \arcsinh(1))$. Let $Z$ be a hyperbolic surface with a single boundary component of length $\varepsilon$, infinite genus, endspace (Ends$(Z)$, Ends$_g(Z)) \simeq (\text{Ends}(S), \text{Ends}_g(S))$ and discrete length spectrum (which exists by Lemma 4.2). By discreteness of $\mathcal{L}(Z)$ and by Remark 4.3, there is $\delta > 0$ such that every closed geodesic and every orthogeodesic in $Z$ has length bigger than $\delta$. We will choose $\delta < \varepsilon$.

Let $V$ be a $(2|G| + 1)$-holed sphere, with boundary components denoted $\{\gamma_g, \bar{\gamma}_g, \gamma \mid g \in G\}$. Fix a hyperbolic structure on $V$ by choosing a pants decomposition $\mathcal{P}$, assigning length via an injective function $\lambda : \mathcal{P} \cup \{\gamma_g, \bar{\gamma}_g \mid g \in G\} \to (0, \delta)$, setting the length of $\gamma$ to be $\varepsilon$ and choosing all twist parameters to be zero.

We then construct a compact hyperbolic surface $F$ modelled on the Cayley graph of $G$ with $G$ as generating set, using $V$ as vertex surface, as in [All06]. Precisely, for every $g$ in $G$, let $V_g$ be a copy of $V$. For any $g, h \in G$, glue the $\gamma_h$ boundary component of $V_g$ to the $\bar{\gamma}_h$ boundary component of $V_{gh}$, with zero twist. As in [All06], the resulting surface has $\text{Isom}(F) = G$, where $G$ acts by rigidly permuting the $V_g$'s. Denote by $\mathcal{Q}$ the collection of curves in $F$ obtained as union of all the curves in $\mathcal{P}$ for every vertex surface $V_g$. Note that these are exactly the simple closed geodesics in $F$ of length at most $\delta$ (by the Collar Lemma).

By construction, $F$ has $|G|$ boundary components, all of length $\varepsilon$ (one per surface $V_g$). For every $g \in G$, glue a copy $Z_g$ of $Z$ to the $\gamma$ boundary component of $V_g$, with zero twist, to get a surface $X$.

<figure><img src="image_3.png" /><figcaption>FIGURE 2. Construction of $X$ for $G = \mathbb{Z}/3\mathbb{Z}$ and $S$ the blooming Cantor tree surface.</figcaption></figure>

The action of $G$ extends to an isometric action on $X$ (by rigidly permuting the $Z_g$), so $G < \text{Isom}(X)$.

Note that the curves in $\mathcal{Q}$ are the only simple closed geodesics in $X$ of length at most $\delta$: indeed, consider a simple closed geodesic $\alpha$. If it is contained in $F$, we have already shown that $\alpha \in \mathcal{Q}$ if and only if its length is less than $\delta$. If it is not contained in $F$, there is a $Z_g$ which either contains $\alpha$
or such that $\alpha \cap Z_g$ contains an arc from the boundary to the boundary. Since all closed geodesics and orthogeodesics of $Z$ have length more than $\delta$, $\alpha$ must have length more than $\delta$.

As a consequence, any isometry $\varphi$ of $X$ needs to send $Q$ to itself. If $\varphi(F)$ were different from $F$, there would be a boundary component of $F$ sent to a curve intersecting the boundary of $F$, which is impossible by the Collar Lemma. So $\varphi$ restricts to an isometry of $F$, and thus agrees on $F$ with some $g \in G$. As two isometries agreeing on an open set are the same map, $\varphi = g$, and hence $\mathrm{Isom}(X) = G$.

By construction $X$ has infinite genus and its endspace is $\sqcup_{g \in G}(\mathrm{Ends}(Z_g), \mathrm{Ends}_g(Z_g))$. As $S$ has self-duplicating endspace, this is homeomorphic to $(\mathrm{Ends}(S), \mathrm{Ends}_g(S))$. So $X$ is homeomorphic to $S$.

Finally, we want to show that $\mathcal{L}(X)$ is discrete. By contradiction, suppose that $\{\alpha_n | n \in \mathbb{N}\}$ is a collection of closed geodesics of uniformly bounded length. By discreteness of $\mathcal{L}(Z)$ and of $\mathcal{L}(F)$, they cannot be all contained in any $Z_g$ or in $F$. If there is a $Z_g$ and a subsequence $n_j$ such that for every $j$, $\alpha_{n_j} \cap Z_g$ contains an arc $a_j$ from the boundary to the boundary of $Z_g$ and the $a_j$ are pairwise not homotopic, by discreteness of the orthospectrum of $Z$ the lengths of the $a_j$ go to infinity, a contradiction. So the only possibility is that all the $\alpha_n$ intersect the $F$ and all the $Z_g$ in the same arcs (up to homotopy relative to the boundary of $F$ or $Z_g$). But then the $\alpha_n$ differ by Dehn twists about the boundary components of $F$, with larger and larger powers (in absolute value). In particular their lengths cannot be uniformly bounded, a contradiction. $\square$

## 5. ARBITRARILY LARGE ISOSPECTRAL FAMILIES

In this section we will prove Theorem B for any infinite-genus surface with self-duplicating endspace, that is:

**Theorem 5.1.** Let $S$ be an infinite-genus surface with self-duplicating endspace. For every $n \in \mathbb{N}$ there is a hyperbolic structure $X$ on $S$ with discrete length spectrum and such that $|\mathcal{I}(X)| \ge n$.

*Proof of Theorem B.* Consider $H = (\mathbb{Z}/8\mathbb{Z})^* \times \mathbb{Z}/8\mathbb{Z}$, $H_1 = \{(1,0), h_1 = (3,0), h_2 = (5,0), (7,0)\}$ and $H_2 = \{(1,0), (3,4), (5,4), (7,0)\}$ as in Section 2. Fix $n \in \mathbb{N}$ and let $G$ be the product of $n$ copies of $H$. Denote by $K_i$, for $i = 1, \dots, n$, the subgroup $H_1 \times \dots H_1 \times H_2 \times \dots \times H_1$, where $H_2$ is at the $i$-th position.

Apply the construction from Theorem C to get a surface $X$ with isometry group $G$, with an extra condition: we require the pants decomposition $\mathcal{P}$ of $V$ to contain a specific set of curves, which we now describe. Define the inclusion of $H$ in the $i$-th component of $G$ as

$$ \begin{array}{l} \iota_i : H \to G \\ h \mapsto (1, \dots, 1, h, \dots, 1). \end{array} $$

Then, for every $i = 1, \dots, n$, at every vertex surface, define three curves in $\mathcal{P}$, $\{\eta_{i,k}\}_{k=1}^3$ such that $\eta_{i,1}$ bounds a pair of pants with $\gamma_{\iota_i(h_1)}$ and $\tilde{\gamma}_{\iota_i(h_1)}$, $\eta_{i,2}$ bounds a pair of pants with $\gamma_{\iota_i(h_2)}$ and $\tilde{\gamma}_{\iota_i(h_2)}$, and $\eta_{i,3}$ bounds a pair of pants with $\eta_{i,2}$ and $\eta_{i,3}$ (see Figure 3).

Consider then the surfaces $X_i = X/K_i$. Our goal is to show that they all belong to $\mathcal{M}(S)$, have discrete length spectrum, are isospectral and pairwise non-isometric. Then $|\mathcal{I}(X_1)| \ge n$ as required.

The proof that all $X_i$ are homeomorphic to $S$ and have discrete spectrum is the same as that for $X$ (in the proof of Theorem C). They are isospectral by Sunada's criterion, i.e. Theorem 1.1. To conclude, we just need to show that they are pairwise not isometric. To this end, suppose $\varphi : X_i \to X_j$, for $i \ne j$, is an isometry. As in the proof of Theorem C, we get that $\varphi$ restricts to
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Construction of $\mathcal{P}$.</figcaption></figure>

an isometry from $F/K_i$ to $F/K_j$. We can then use our choice of $\mathcal{P}$ and the same argument as in [Fan23, Section 5.1] to show that the surfaces are not isometric, concluding the proof.

<figure><img src="image_5.png" /></figure>

## 6. LIMITS OF ISOSPECTRAL FAMILIES

One of the reasons why discreteness of the length spectrum is an interesting assumption is that it is one of the crucial ingredient in Wolpert’s proof of finiteness of isospectral families in the closed case [Wol79]. The sketch of the proof is the following: given a sequence of isospectral surfaces in a moduli space, note that it sits in a compact set — by Mumford’s criterion [Mum71], since all surfaces in the sequence have the same systole. So we can extract a converging subsequence, and lift it to a converging subsequence in Teichmüller space. By discreteness of the length spectrum, and using the fact that the lengths of finitely many curves determine a point in Teichmüller space (see e.g. the 9g–9 theorem [FM12, Theorem 10.7]), Wolpert then shows that the sequence is eventually constant. In particular, any limit of isospectral surfaces is isospectral to all surfaces in the sequence.

As discussed, in the infinite-type setting infinite isospectral families exist, and indeed the main tools used by Wolpert (Mumford’s criterion, the 9g–9 theorem and discreteness of the length spectrum) are missing in this setup. In attendance of possible future constructions of infinite families of discrete isospectral structures, if we do assume discreteness, we can try and see which partial results can be recovered. We end this article by showing that discreteness allows us to show that the limit of isospectral surfaces also has the same spectrum.

**Proposition 6.1.** Let $X$ be a hyperbolic surface with discrete length spectrum. Suppose $Y_k \in \mathcal{I}(X)$ is a converging sequence, with limit $Y$. Then $Y \in \mathcal{I}(X)$.

To prove this, we will use the following lemma:

**Lemma 6.2.** Suppose $\mathcal{L}(X)$ is discrete and $Y_k \in \tilde{\mathcal{I}}(X)$ is a converging sequence, with limit $Y$. Let $L > 0$. Then there is $k_0 \ge 1$ so that for every curve $\gamma$ with $\ell_Y(\gamma) \le L$ and for every $k \ge k_0$, $\ell_{Y_k}(\gamma) = \ell_Y(\gamma)$.

*Proof.* Fix $L$; by discreteness, there is $\varepsilon > 0$, which we assume smaller than $L$, such that every interval $I \subset [0, 2L]$ of length at most $\varepsilon$ contains at most one value of $\mathcal{L}(X)$.

Since $Y_k$ converges, there is $k_0$ so that for every $k \ge k_0$, $\ell_{Y_k}(\gamma) \in (\ell - \varepsilon/2, \ell + \varepsilon/2) \cap [0, 2L]$, where $\ell = \ell_Y(Y_k)$. But then by the previous observation $\ell_{Y_k}(\gamma)$ is the unique element of $\mathcal{L}(X)$ in the given
interval, and thus the sequence is constant for $k \ge k_0$. Since $Y_k \to Y$, the constant can only be $\ell_Y(\gamma)$. $\square$

*Proof of Proposition 6.1.* Lift the sequence $Y_k$ and the limiting point $Y$ to a sequence in $\tilde{\mathcal{I}}(X)$ and limiting point in Teichmüller space. Abusing notation, we call the lifted surfaces $Y_k$ and $Y$.

Let $\ell \in \mathcal{L}(Y)$. Then there is $\gamma$ with $\ell_Y(\gamma) = \ell$, and by the proposition $\ell_{Y_k}(\gamma)$ is eventually constant and equal to $\ell$. Thus $\ell \in \mathcal{L}(Y_k) = \mathcal{L}(X)$ for every sufficiently large $k$. In particular, the set of lengths (forgetting multiplicities) of closed geodesics in $Y$ is contained in that of $X$. To show isospectrality it is then enough to show that each $\ell \in \mathcal{L}(X)$ appears in $\mathcal{L}(Y)$ with the correct multiplicity.

Consider $\ell \in \mathcal{L}(X)$, and suppose its multiplicity is $m > 0$. For every $k$, there are exactly $m$ curves $\gamma_1^k, \dots, \gamma_m^k$ such that

$$ \ell_{Y_k}(\gamma_i^k) = \ell $$

for every $i$. Note that, for every sufficiently large $k$, $\ell_Y(\gamma_i^k) \le 2\ell =: L$. Let $k_0$ be given by the lemma; then for every $k \ge 0$

$$ \ell_Y(\gamma_i^{k_0}) = \ell_{Y_{k_0}}(\gamma_i^{k_0}) = \ell_{Y_k}(\gamma_i^{k_0}) = \ell $$

for every $i = 1, \dots, m$. So $\ell \in \mathcal{L}(Y)$ with multiplicity at least $m$, and moreover (up to renumbering)

$$ \gamma_i^k = \gamma_i^{k_0} $$

for every $k \ge k_0$. So $\ell \in \mathcal{L}(Y)$ and has multiplicity at least $m$.

Now suppose that $\gamma$ is a curve with $\ell_Y(\gamma) = \ell$. By the lemma, $\ell_{Y_k}(\gamma)$ is eventually constant and equal to $\ell_Y(\gamma) = \ell$. So $\gamma$ is one of the $\{\gamma_i^{k_0} | i = 1, \dots, m\}$. This implies that $\ell$ has multiplicity $m$ in $\mathcal{L}(Y)$. $\square$

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Email address: federica.fanoni@u-pec.fr

CNRS, UNIV PARIS EST CRETEIL, UNIV GUSTAVE EIFFEL, LAMA UMR8050, F-94010 CRETEIL, FRANCE

Email address: david.fisac-camara@cnrs.fr

UNIV PARIS EST CRETEIL, UNIV GUSTAVE EIFFEL, CNRS, LAMA UMR8050, F-94010 CRETEIL, FRANCE