Counting Essential Surfaces in a Closed Hyperbolic 3-Manifold

Counting Mapping Class group orbits on hyperbolic surfaces

Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[γ]$ of a closed curve $γ\in π_{1}(S_{g,n})$ contains a unique closed geodesic on $X$. Let $\ell_γ(X)$ denote the hyperbolic length of the geodesic representative of $γ$ on $X$. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on $S_{g,n}.$ As an application, one can obtain the asymptotics of the growth of $s^{k}_{X}(L)$, the number of closed curves of length $\leq L$ on $X$ with at mo…

Counting subgraphs in hyperbolic graphs with symmetry

This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on hyperbolic groups to the hyperbolike setting, including rationality of generating functions, and sharp estimates on the growth rate of vertices. We then apply these results to confirm a conjecture of Saito on the "opposite series", which was originally posed for hyperbolic groups.

Counting immersed surfaces in hyperbolic 3-manifolds

We count the number of conjugacy classes of maximal, genus g, surface subroups in hyperbolic 3-manifold groups. For any closed hyperbolic 3-manifold, we show that there is an upper bound on this number which grows factorially with g. We also give a class of closed hyperbolic 3-manifolds for which there is a lower bound of the same type.

Effective counting of simple closed geodesics on hyperbolic surfaces

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmüller geodesic flow.

Counting problems in graph products and relatively hyperbolic groups

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including right-angled Artin groups and right-angled Coxeter groups.

Counting Curves in Hyperbolic Surfaces

Let $Σ$ be a hyperbolic surface. We study the set of curves on $Σ$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $γ_0$. For example, in the particular case that $Σ$ is a once-punctured torus, we prove that the cardinality of the set of curves of type $γ_0$ and of at most length $L$ is asymptotic to $L^2$ times a constant.

Counting hyperbolic multi-geodesics with respect to the lengths of individual components

Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $\leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L \to \infty$ to a polynomial in $L$ of degree $6g-6+2n$. We establish asymptotics of the same kind for countings of multi-geodesics in mapping class group orbits of simple or filling closed multi-curves that keep track of the hyperbolic lengths of individual components, proving and generalizing a…

Mineyev流的混合性、轨道计数与强双曲度量下的Poincaré级数

Mixing of the Mineyev flow, orbital counting and Poincaré series for strongly hyperbolic metrics

本研究结合遍历理论与符号动力学，为双曲群上的强双曲度量建立了轨道计数结果，并证明了Mineyev拓扑流的混合性及Poincaré级数的解析域。

Counting cusped hyperbolic 3-manifolds that bound geometrically

We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.

Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics

We show that the number of square-tiled surfaces of genus $g$, with $n$ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $L$ squares, is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.

闭合双曲面拼接的计数与熵

Counting and entropy for hyperbolic surface amalgams

研究闭合双曲面拼接中闭合测地线数量的增长，证明拓扑熵与体积熵相等，但其行为受几何数据影响，特别是通过系统孔径长度和拼接曲线长度的上下界影响，熵在无系统孔径下界时可指数增长。

将伪阿诺索夫映射类计数为弱收缩等距映射

Counting pseudo-Anosovs as weakly contracting isometries

本文证明在有限型双曲曲面映射类群及更广的层级双曲群的Cayley图中，伪阿诺索夫映射类或Morse元素是普遍的，并建立了拟等距不变量计数理论。

Hecke群中倒数双曲元素的计数

Counting Reciprocal Hyperbolic Elements in Hecke Groups

本文研究Hecke群中倒数双曲元素的计数，确定了其素共轭类数量的渐近增长率与极限常数。

Infinitely many meridional essential surfaces of bounded genus in hyperbolic knot exteriors

We show the existence of an infinite collection of hyperbolic knots where each of which has in its exterior meridional essential planar surfaces of arbitrarily large number of boundary components, or, equivalently, that each of these knots has essential tangle decompositions of arbitrarily large number of strings. Moreover, each of these knots has in its exterior meridional essential surfaces of any positive genus and (even) number of boundary components. That is, the compact surfaces that have a meridional essential embedding into a hyperbolic knot exterior have meridional essential embedding…

Counting curve types

Let $S$ be a closed orientable hyperbolic surface, and let $\mathcal{O}(K,S)$ denote the number of mapping class group orbits of curves on $S$ with at most $K$ self-intersections. Building on work of Sapir [16], we give upper and lower bounds for $\mathcal{O}(K,S)$ which are both exponential in $\sqrt{K}$.

Counting loxodromics for hyperbolic actions

Let $G \curvearrowright X$ be a nonelementary action by isometries of a hyperbolic group $G$ on a hyperbolic metric space $X$. We show that the set of elements of $G$ which act as loxodromic isometries of $X$ is generic. That is, for any finite generating set of $G$, the proportion of $X$--loxodromics in the ball of radius $n$ about the identity in $G$ approaches $1$ as $n \to \infty$. We also establish several results about the behavior in $X$ of the images of typical geodesic rays in $G$; for example, we prove that they make linear progress in $X$ and converge to the Gromov boundary $\partia…

Pseudo-Developing Maps for Ideal Triangulations I: Essential Edges and Generalised Hyperbolic Gluing Equations

Let N be a topologically finite, orientable 3-manifold with ideal triangulation. We show that if there is a solution to the hyperbolic gluing equations, then all edges in the triangulation are essential. This result is extended to a generalisation of the hyperbolic gluing equations, which enables the construction of hyperbolic cone-manifold structures on N with singular locus contained in the 1-skeleton of the triangulation.

Counting essential surfaces in 3-manifolds

We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected) essential surfaces in terms of their Euler characteristic always has a short generating function and hence has quasi-polynomial behavior. This gives remarkably concise formulae for the number of such surfaces, as well as detailed asymptotics. We give algorithms that allow us to compute these generating functions and the underlying surfaces, and apply these …

The smallest Haken hyperbolic polyhedra

We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3-orbifold contains an essential 2-suborbifold, up to a canonical decomposition along essential hyperbolic triangle 2-suborbifolds.

Counting essential minimal surfaces in closed negatively curved n-manifolds

For closed odd-dimensional manifolds with sectional curvature less or equal than -1, we define the minimal surface entropy that counts the number of surface subgroups. It attains the minimum if and only if the metric is hyperbolic. Moreover, we compute the entropy associated with other locally symmetric spaces and cusped hyperbolic 3-manifolds.

A stochastic approach to counting problems

We study orbital functions associated to finitely generated geometrically infinite Kleinian groups acting on the hyperbolic space $\mathbb{H}^3$, developing a new method based on the use of the Brownian motion. On the way, we give some estimates of the orbital function associated to nilpotent covers of compact hyperbolic manifolds, partially answering a question asked by M. Pollicott to the author.

计数具有凹口的双曲3流形中的曲面子群

Counting surface subgroups in cusped hyperbolic 3-manifolds

该研究证明了双曲3流形中曲面子群的数量随曲面亏格增长的下界，并构造了具有特定性质的曲面子群示例。

Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves

In her thesis, Mirzakhani showed that the number of simple closed geodesics of length $\leq L$ on a closed, connected, oriented hyperbolic surface $X$ of genus $g$ is asymptotic to $L^{6g-6}$ times a constant depending on the geometry of $X$. In this survey we give a detailed account of Mirzakhani's proof of this result aimed at non-experts. We draw inspiration from classic primitive lattice point counting results in homogeneous dynamics. The focus is on understanding how the general principles that drive the proof in the case of lattices also apply in the setting of hyperbolic surfaces.

通过德恩手术得到双曲三维流形的一般性

Genericity of hyperbolic 3-manifolds via Dehn surgery

研究如何描述链环和德恩手术性质，引入计数模型，证明在该模型下随机链环和随机三维流形是双曲的。

Prime number theorems and holonomies for hyperbolic rational maps

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both estimates have power savings error terms. Our counting and equidistribution results will follow from a study of dynamical zeta functions that have been twisted by characters of $S^1$. We will show that these zeta functions are non-vanishing on a half plane $\Re(s) > δ- ε$, where $…

Counting Salem numbers of arithmetic hyperbolic 3-orbifolds

It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic $3$-dimensional orbifold defines $c Q^{1/2} + O(Q^{1/4})$ square-rootable Salem numbers of degree $4$ which are less than or equal to $Q$. This quantity can be compared to the total number of such Salem numbers, which is shown to be asymptotic to $\frac{4}{3}Q^{3/2}+O(Q)$. Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic $3$-orbifolds. As an applic…

Toroidal Dehn fillings on large hyperbolic 3-manifolds

We show that if a hyperbolic 3-manifold $M$ with a single torus boundary admits two Dehn fillings at distance 5, each of which contains an essential torus, then $M$ is a rational homology solid torus, which is not large in the sense of Wu. Moreover, one of the surgered manifold contains an essential torus which meets the core of the attached solid torus minimally in at most two points. This completes the determination of best possible upper bounds for the distance between two exceptional Dehn fillings yielding essential small surfaces in all ten cases for large hyperbolic 3-manifolds.

计数来自算术双曲轨道的Salem数

Counting Salem numbers arising from arithmetic hyperbolic orbifolds

本文研究算术双曲轨道中Salem数与短测地线的关系，通过界定了Salem数比例和测地线谱乘法的指数增长下界，改进了相关理论。

Essential Surfaces in the Exterior of K13n586

We count the number of isotopy classes of closed, connected, orientable, essential surfaces embedded in the exterior B of the knot K13n586.The main result is that the count of surfaces by genus is equal to the Euler totent function. This is the first manifold for which we know the number of surfaces for any genus. The main argument is to show when normal surfaces in B are connected by counting their number of components. We implement tools from Agol, Hass and Thurston to convert the problem of counting components of surfaces into counting the number of orbits in a set of integers under a colle…

计数双陪集及其在一般三维流形中的应用

Counting double cosets with application to generic 3-manifolds

本研究推广了双陪集增长理论至更广群类，证明 Morse 子群双陪集增长与轨道增长相当，并应用于证实 Maher 猜想：双曲三维流形在 Heegaard 分解流形中指数泛型。

Equidistribution and Counting for orbits of geometrically finite hyperbolic groups

Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Gamma discrete, we compute an asymptotic formula for the number of points in w_0Gamma of norm at most T, provided that the Bowen-Margulis-Sullivan measure on the associated hyperbolic manifold and the Gamma skinning size of w_0 are finite. The main ergodic ingredient in our approac…

Embedding of metric graphs on hyperbolic surfaces

An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_e(G)$ of $(G, d)$ is the lowest genus of a surface on which such an embedding is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ admits such an embedding (…

On the minimal diameter of closed hyperbolic surfaces

We prove that the minimal diameter of a hyperbolic compact orientable surface of genus $g$ is asymptotic to $\log g$ as $g \to \infty$. The proof relies on a random construction, which we analyse using lattice point counting theory and the exploration of random trivalent graphs.

Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond

We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schottky dances, etc.

Handle additions producing essential surfaces

We construct a small, hyperbolic 3-manifold $M$ such that, for any integer $g\geq 2$, there are infinitely many separating slopes $r$ in $\partial M$ so that $M(r)$, the 3-manifold obtained by attaching a 2-handle to $M$ along $r$, is hyperbolic and contains an essential separating closed surface of genus $g$. The result contrasts sharply with those known finiteness results on Dehn filling, and it also contrasts sharply with the known finiteness result on handle addition for the cases $g=0,1$. Our 3-manifold $M$ is the complement of a hyperbolic, small knot in a handlebody of genus 3.

Depth of pleated surfaces in toroidal cusps of hyperbolic 3-manifolds

Let $F$ be a closed essential surface in a hyperbolic 3-manifold $M$ with a toroidal cusp $N$. The depth of $F$ in $N$ is the maximal distance from points of $F$ in $N$ to the boundary of $N$. It will be shown that if $F$ is an essential pleated surface which is not coannular to the boundary torus of $N$ then the depth of $F$ in $N$ is bounded above by a constant depending only on the genus of $F$. The result is used to show that an immersed closed essential surface in $M$ which is not coannular to the torus boundary components of $M$ will remain essential in the Dehn filling manifold $M(γ)$ a…

Counting minimal surfaces in negatively curved 3-manifolds

We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.

Laminar free hyperbolic 3-manifolds

We analyse the existence question for essential laminations in 3-manifolds. The purpose is to prove that there are infinitely many closed hyperbolic 3-manifolds which do not admit essential laminations. This answers in the negative a question posed by Gabai and Oertel. The proof is obtained by analysing certain group actions on trees and showing that certain 3-manifold groups only have trivial actions on trees. In general the trees are neither simplicial nor metric. There are corollaries concerning the existence question for Reebless foliations and pseudo-Anosov flows.

Dehn filling of cusped hyperbolic 3-manifolds with geodesic boundary

We define for each g>=2 and k>=0 a set M_{g,k} of orientable hyperbolic 3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of genus g. Manifolds in M_{g,k} have Matveev complexity g+k and Heegaard genus g+1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In addition, they do not contain closed essential surfaces. The cardinality of M_{g,k} for a fixed k has growth type g^g. We completely describe the non-hyperbolic Dehn fillings of each M in M_{g,k}, showing that, on any cusp of any hyperbolic manifold obtained by partially filling M, the…

Central limit theorems for counting measures in coarse negative curvature

We establish central limit theorems for an action of a group G on a hyperbolic space X with respect to the counting measure on a Cayley graph of G. Our techniques allow us to remove the usual assumptions of properness and smoothness of the space, or cocompactness of the action. We provide several applications which require our general framework, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds.

Finite Groups and Hyperbolic Manifolds

The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n > 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n = 3 have been proven by Greenberg and Kojima, respectively. Our proof is non constructive: it uses counting results from subgroup growth theory and the strong approximation theorem to show that such manifolds exist.

A norm on homology of surfaces and counting simple geodesics

We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.

Thurston's Spinning Construction and Solutions to the Hyperbolic Gluing Equations for Closed Hyperbolic 3-Manifolds

We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we show that this gives a new algorithm to detect hyperbolic structures on closed 3-manifolds.

Counting one sided simple closed geodesics on Fuchsian thrice punctured projective planes

We prove that there is a true asymptotic formula for the number of one sided simple closed curves of length $\leq L$ on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic structure, and it is noninteger, in contrast to counting results of Mirzakhani for simple closed curves on orientable Fuchsian surfaces.

The shapes of complementary subsurfaces to simple closed hyperbolic multi-geodesics

Cutting a hyperbolic surface X along a simple closed multi-geodesic results in a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to infinity, showing that they equidistribute to the Kontsevich measure on a corresponding moduli space of metric ribbon graphs. In particular, random subsurfaces look like random ribbon graphs, a law which does not depend on the initial choice of X. This result strengthens Mirzakhani's famous simple closed multi-geodesic counting theorems for hyperbolic surfaces…

Reducing Dehn fillings and small surfaces

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. Especially we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essential sphere, projective plane, annulus, Moebius band, torus or Klein bottle,all 11 pairs of such non-hyperbolic manifolds.

On the complexity of cusped non-hyperbolicity

We show that the problem of showing that a cusped 3-manifold M is not hyperbolic is in NP, assuming $S^3$-RECOGNITION is in coNP. To this end, we show that IRREDUCIBLE TOROIDAL RECOGNITION lies in NP. Along the way we unconditionally recover SATELLITE KNOT RECOGNITION lying in NP. This was previously known only assuming the Generalized Riemann Hypothesis. Our key contribution is to certify closed essential normal surfaces as essential in polynomial time in compact orientable irreducible $\partial$-irreducible triangulations. Our work is made possible by recent work of Lackenby showing several …

Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length th…

计算给定换位子长度的测地线

Counting geodesics of given commutator length

本文研究闭双曲曲面上可表示为固定数目换位子乘积的周期测地线计数渐近，并通过计数三价图临界实现的方法加以解决，还据此证明 Huber 几何素数定理。

关于5到12维右角双曲多面体理想顶点和有限顶点数量下限的研究

On lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra in dimensions from 5 to 12

研究通过几何方法确定了5维右角双曲多面体至少有3个理想顶点，7维至少有4个有限顶点，并通过递推关系扩展到12维。

克莱因群强收敛的一致谱隙与正交测地线计数

Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups

本文研究克莱因群强收敛下几何有限双曲流形的小特征值收敛性，利用极限流形的谱隙与测地流指数混合性质，建立了凸集间正交测地线数量的渐近一致计数公式，并应用于 Margulis 管、测地环及本原闭测地线。

Square-integrability of the Mirzakhani function and statistics of simple closed geodesics on hyperbolic surfaces

Given integers $g,n \geq 0$ satisfying $2-2g-n < 0$, let $\mathcal{M}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B \colon \mathcal{M}_{g,n} \to \mathbf{R}_{\geq 0}$ which assigns to $X \in \mathcal{M}_{g,n}$ the Thurston measure of the set of measured geodesic laminations on $X$ of hyperbolic length $\leq 1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $\mathcal{M}_{g,n}$ and deduce that $B$ is square-integrable wi…

Counting Conjugacy Classes in $Out(F_N)$

We show that if a f.g. group $G$ has a non-elementary WPD action on a hyperbolic metric space $X$, then the number of $G$-conjugacy classes of $X$-loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$. As an application we prove that for $N\ge 3$ the number of distinct $Out(F_N)$-conjugacy classes of fully irreducibles $ϕ$ from an $R$-ball in the Cayley graph of $Out(F_N)$ with $\logλ(ϕ)$ on the order of $R$ grows exponentially in $R$.

填充链环与本质收缩

Filling Links and Essential Systole

本文回答了Freedman和Krushkal的问题，在任意闭可定向3-流形中构造了具有大本质收缩的双曲填充链环，突破了先前对双曲链环的几何限制。