Core surfaces

Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq 2$ . We develop a new method for integrating over the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$ . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g. Given $\phi \in \mathbb {X}_{g,n}$ and $\gamma \in \Gamma _{g}$ , we let $\mathsf {fix}_{\gamma }(\phi )$ be the number of fixed points of the permutation $\phi (\gamma )$ . The function $\mathsf {fix}_{\gamma }$ is a special case of a natural family of functions on $\mathbb {X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $n\to \infty $ , for the expectation of $\mathsf {fix}_{\gamma }$ with respect to the uniform probability measure on $\mathbb {X}_{g,n}$ , which is denoted by $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ . We prove that if $\gamma \in \Gamma _{g}$ is not the identity and q is maximal such that $\gamma $ is a qth power in $\Gamma _{g}$ , then $$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=d(q)+O(n^{-1}) \end{align*}$$ as $n\to \infty $ , where $d\left (q\right )$ is the number of divisors of q. Even the weaker corollary that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]=o(n)$ as $n\to \infty $ is a new result of this paper. We also prove that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$ .

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“…[MP22a, Def. 3.1] A tiled surface Y is a subcomplex of a (not-necessarily-connected) covering space of .…”

Section: Resolutions Of Core Surfacesmentioning

confidence: 99%

“…The first observation is that , the expected number of fixed points of under , is the expected number of times that we see a fixed annulus A , specified by , immersed in the random tiled surface . This annulus A may be the ‘core surface’ corresponding to – see Definition 2.6, the left part of Figure 2.3 and [MP22a, Lem. 5.1].…”

Section: Introductionmentioning

confidence: 99%

The Asymptotic Statistics of Random Covering Surfaces

Magee

Puder

2023

Forum of Mathematics, Pi

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“…Call the resulting complex C γ . That C γ is a tiled surface follows from [MP21] (in particular, it is embedded in the core surface Core ( γ ) which is itself a tiled surface, by [MP21, Thm. 5.10]).…”

Section: We Let ∂Y Denote the Boundary Of The Thick Version Of Y And ...mentioning

confidence: 99%

A random cover of a compact hyperbolic surface has relative spectral gap $$\frac{3}{16}-\varepsilon $$

2022

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Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature $$-1$$ - 1 . For each $$n\in {\mathbf {N}}$$ n ∈ N , let $$X_{n}$$ X n be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or $$X_{n}$$ X n is an eigenvalue of the associated Laplacian operator $$\Delta _{X}$$ Δ X or $$\Delta _{X_{n}}$$ Δ X n . We say that an eigenvalue of $$X_{n}$$ X n is new if it occurs with greater multiplicity than in X. We prove that for any $$\varepsilon >0$$ ε > 0 , with probability tending to 1 as $$n\rightarrow \infty $$ n → ∞ , there are no new eigenvalues of $$X_{n}$$ X n below $$\frac{3}{16}-\varepsilon $$ 3 16 - ε . We conjecture that the same result holds with $$\frac{3}{16}$$ 3 16 replaced by $$\frac{1}{4}$$ 1 4 .

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“…As we saw in Theorem 2.11, any group with a given presentation can be represented by a 2-dimensional CW complex. The idea of "core surfaces" developed in [10] opens the door to proving results about more general classes of groups, such as finite groups, abelian…”

Section: Discussionmentioning

confidence: 99%

“…Even more recently, Jordi Delgado and Enric Ventura apply this approach to the study of direct products of finitely generated free and abelian groups in Stallings automata for free-times-abelian groups: intersections and index [9]. In Core Surfaces [10], Michael Magee and Doron Puder generalize Stallings' theory of core graphs to fundamental groups of surfaces (usually 2-dimensional CW complexes) and their subgroups. The application of this theory to so many different classes of groups suggests a rich opportunity for further research in group theory.…”

Section: Introductionmentioning

confidence: 99%

On Applications of Topological and Combinatorial Methods to the Theory of Groups

Santean¹

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Graphs are topological objects called 1-dimensional CW complexes, and their fundamental groups are free groups. More generally, any group can be represented by a 2-dimensional CW complex, which is a graph with discs glued along the boundaries of closed paths corresponding to relations in the group. These objects can be studied from the topological viewpoint of covering space theory, introduced by John R. Stallings, which allows us to visualize groups and determine their subgroup structure. Alternatively, graphs can be studied from a combinatorial point of view, developed by Ilya Kapovich and Alexei Myasnikov, which provides simple algorithms that answer questions about free groups. We give an exposition of both approaches and demonstrate how they are used to answer questions about subgroups of free groups and free products.

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Core surfaces

Cited by 3 publications

References 14 publications

The Asymptotic Statistics of Random Covering Surfaces

The Asymptotic Statistics of Random Covering Surfaces

A random cover of a compact hyperbolic surface has relative spectral gap $$\frac{3}{16}-\varepsilon $$

On Applications of Topological and Combinatorial Methods to the Theory of Groups

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