Organizers: Spencer Dowdall, Dan Margalit, Denis Osin

Time: Wednesdays, 4:10–5:00pm

Location: SC 1432

Mailing List: If you would like to be added to the seminar mailing list and receive regular announcements, please contact an organizer.

Fall 2023

Wednesday, September 6

Spencer Dowdall (Vanderbilt)

Title: Counting finite-order elements of mapping class groups

Abstract: I will discuss the growth rate of the number of finite-order elements of the mapping class group of a surface S of genus g. We approach this question from the perspective of the so-called lattice point counting problem for the mapping class group acting on Teichmuller space, which concerns the number of group elements that send a given point into a ball of radius R about another point. Athreya, Bufetov, Eskin, and Mirzakhani have shown that, for the whole mapping class group, this quantity is asymptotic to e(6g-6)R as R tends to infinity. We instead consider only orbit points that are translates by finite-order elements and show that the associated count grows at the rate of e(3g-3)R, that is, with exactly half the exponent. In order to achieve this, we introduce a new notion of "complexity length" in Teichmuller space which has several interesting features reflecting aspects of negative curvature. Joint work with Howard Masur.

Wednesday, September 13

Sam Sheferd (Vanderbilt)

Title: Graphically discrete groups

Abstract: I will introduce the notion of a group G being graphically discrete, which relates to the discreteness of automorphism groups of graphs that admit geometric G-actions. I will discuss the definition along with examples and non-examples, and how it can be used to prove rigidity results.

Wednesday, September 20

Koichi Oyakawa (Vanderbilt)

Title: Hyperfiniteness of boundary actions of acylindrically hyperbolic groups

Abstract: A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure complexity of Borel equivalence relations. Although group actions on hyperbolic spaces don't always induce hyperfinite orbit equivalence relations on the Gromov boundary, some natural boundary actions were recently found to be hyperfinite. Examples of such actions include actions of hyperbolic groups and relatively hyperbolic groups on their Gromov boundary and acylindrical group actions on trees. In this talk, I will show that any acylindrically hyperbolic group admits a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action.

Wednesday, September 27

Bin Sun (Michigan State University)

Title: Outer automorphisms of acylindrically hyperbolic groups with Kazhdan’s property (T)

Abstract: The combination of Kazhdan's property (T) and negative curvature typically limits the number of outer automorphisms. Indeed, it is a result of Paulin that every property (T) hyperbolic group has a finite outer automorphism group. Analogous results have been proved for relatively hyperbolic groups by Belegradek-Szczepanski and lacunary hyperbolic groups by Coulon-Guirardel. We proved that this analogy does not carry over to acylindrically hyperbolic groups. Specifically, we showed that, for every countable group Q, there is a property (T) acylindrically hyperbolic group G such that Out(G) is isomorphic to Q. We also proved the following converse to Paulin’s result: for every finite group Q, there is a property (T) hyperbolic group G such that Out(G) is isomorphic to Q. This is joint work with Ionut Chifan, Adrian Ioana, and Denis Osin.

Wednesday, October 4

Srivatsav Kunnawalkam Elayavalli (UCSD)

Title: Representation growth of property (T) groups and Voiculescu's entropy theory.

Abstract: I will describe a connection between an old difficult question of Wigderson from the 60's on representation growth of countable property (T) groups and 1-bounded entropy which is a modern invariant of von Neumann algebras introduced by Jung and Hayes. It seems to be a win-win situation: either there is a negative solution to Wigderson's question or there is a property (T) von Neumann algebra with 1-bounded entropy strictly between 0 and infinity, which would be a very powerful tool in von Neumann algebra theory that would bring new light into old open problems in II1-factor theory. Moreover, under the assumption of flexible Hilbert-Schmidt stability, these two problems are shown to be the same. The talk will be accessible to group theorists, I will not get into von Neumann algebra theory. This is based on joint work with Hayes and Jekel, and Hayes and Thom.

Wednesday, October 11

Bena Tshishiku (Brown University)

Title: Nielsen realization for 3-manifolds

Abstract: Given a manifold M, the Nielsen realization problem asks when a finite subgroup of the mapping class group Mod(M) lifts to the diffeomorphism group Diff(M) under the natural projection Diff(M) → Mod(M). In this talk, we consider the Nielsen realization problem for 3-manifolds and give a solution for subgroups of Mod(M) generated by sphere twists. This is joint work with Lei Chen.

Wednesday, October 18

Denis Osin (Vanderbilt)

Title: Out(Fn)-invariant measures on the space of finitely generated marked groups.

Abstract: Let Gn denote the space of n-generated marked groups. The natural action of the group Out(Fn) on Gn gives rise to a topological dynamical system, which is of fundamental importance to the study of the isomorphism relation on Gn. The goal of my talk is to discuss various questions related to the existence of an invariant measure for this action. In particular, I will utilize acylindrical hyperbolicity of Aut(Fn) to show that Gn admits a non-atomic, ergodic, Out(Fn)-invariant probability measure. This answers a question of Grigorchuk. I will also discuss some model-theoretic implications of this result.

Wednesday, October 25

Jone Lopez de Gamiz Zearra (Vanderbilt)

Title: Separability properties of higher-rank GBS groups

Abstract: A rank n generalized Baumslag-Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to free abelian groups of rank n. This class of groups extends the class of (generalized) Baumslag-Solitar groups. In this talk, we first present what has been known about residual finiteness and subgroup separability for certain subclasses of this class. Then we completely classify these groups in terms of their separability properties. Specifically, we determine when they are residually finite, subgroup separable, and cyclic subgroup separable.

Wednesday, November 1

Aidan Lorenz (Vanderbilt)

Title: Fibered 3-manifolds and minimal dilatation pseudo-Anosov homeomorphisms

Abstract: A result of Thurston’s says that a fibered 3-manifold is hyperbolic if and only if it has a pseudo-Anosov monodromy. Moreover, if the second homology of the 3-manifold has dimension larger than or equal to 2, there are infinitely many different ways in which the manifold fibers and hence infinitely many pseudo-Anosov monodromies. Thus one can study pseudo-Anosov homeomorphisms and in particular their dilatations by way of hyperbolic fibered 3-manifolds. In this talk we will discuss techniques to do so and some related results about minimal dilatation pseudo-Anosovs.

Wednesday, November 8

Fedya Manin (UCSB)

Title: Degrees of maps and multiscale geometry

Abstract: I will talk about the following question of Gromov: what closed manifolds can be efficiently wrapped with Euclidean wrapping paper? That is, for what M is there a 1-Lipschitz map Rn→ M with positive asymptotic degree? Gromov called such manifolds elliptic. We show that, for example, the connected sum of k copies of CP2 is elliptic if and only if k ≤ 3. I will try to explain the intuition behind this example, how it extends to a more general dichotomy governed by the de Rham cohomology of M, and why ellipticity is central to the program of understanding the relationship between topology and metric properties of maps. If I have time, I'll also explain why for a non-elliptic M, a maximally efficient map Rn→ M must have components at many different frequencies (in a Fourier-analytic sense), and even then it's at best logarithmically far from having positive asymptotic degree. This is joint work with Sasha Berdnikov and Larry Guth.

Wednesday, November 15

Yulan Qing (University of Toronto)

Title: Geometric Boundary of Groups

Abstract: Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it is an essential tool in the study of the coarse geometry of hyperbolic groups. In this talk, we generalize the Gromov boundary. We first construct the sublinearly Morse boundaries and show that it is a QI-invariant, metrizable topological space. We show sublinearly Morse directions are generic both in the sense of Patterson-Sullivan and in the sense of random walk. The sublinearly Morse boundary is a subset of all directions with desired properties. In the second half we will truly consider the space of all directions and show that with some minimal assumptions on the space, the resulting boundary is a QI-invariant topology space in which many existing boundaries embed. This talk is based on a series of work with Kasra Rafi and Giulio Tiozzo.

Wednesday, November 29

Yandi Wu (UW Madison)

Title: Marked Length Spectrum Rigidity for Surface Amalgams

Abstract: The marked length spectrum of a negatively curved metric space can be thought of as a length assignment to every closed geodesic in the space. A celebrated result by Croke and Otal says that metrics on negatively curved closed surfaces are determined completely by their marked length spectra. Apart from being natural generalizations of surfaces, surface amalgams also have fundamental groups that provide examples of limit groups and finite-index subgroups of RACGs. In my talk, I will discuss my work towards extending Croke and Otal’s result to a large class of surface amalgams.

Wednesday, December 6

Kevin Boucher (University of Southampton)

Title: Uniformly bounded representations of hyperbolic groups.

Abstract: After an introduction to the subject of boundary representations of hyperbolic groups, I will present some recent developments motivated by a spectral formulation of the so-called Shalom conjecture. This is a joint work with Dr Jan Spakula.

Wednesday, December 13

Öykü Yurttaş (Dicle University)

Title: Fast computations in mapping class groups via global coordinates

Abstract: I will discuss joint work with Margalit, Strenner, and Taylor that gives a quadratic-time algorithm to compute the stretch factor and the invariant foliations for a pseudo-Anosov element of the mapping class group. Our algorithm can efficiently be realized by making use of global coordinates on the boundary of Teichmüller space. After describing our algorithm, I will discuss joint work with Alessandrini in which we provide global coordinates for punctured surfaces, and give explicit formulae for the action of each generator of the mapping class group in terms of these coordinates.

Spring 2024

Wednesday, January 17

Dan Margalit (Vanderbilt University)

Title: Mapping class groups in complex dynamics

Abstract: A rational function in one variable can be regarded as a map from the Riemann sphere to itself. One way to study such a function is through its topology, that is, its homotopy class (relative to a finite set of points). To do this, we can use many of the tools from the theory of mapping class groups, the theory of homotopy classes of surface homeomorphisms. In this talk, I'll explain this connection, and describe how to classify rational functions in a manner analogous to the way we classify surface homeomorphisms.

Wednesday, January 24

Karim Sane (Cheikh Anta Diop University)

Title: Connected components of surgery graph on unicellular graphs

Abstract: A unicellular graph is the isotopy class of a graph embedded in a closed surface whose complement is a disk. There is an operation called surgery that turns a unicellular graph into another one. This endows the set of unicellular graphs with a graph structure called a surgery graph: the vertices are unicellular graphs, and the edges are given by surgeries. In this talk, we will discuss the connectedness of surgery graphs and their relation to the mapping class group and its subgroups (one of which is the liftable mapping class group). This is joint work with Nick Salter.

Wednesday, January 31

Hannah Hoganson (University of Maryland)

Title: A Geometric "Big Out(Fn)" and its Coarse Geometry

Abstract: There are many parallels in the studies of mapping class groups and outer automorphism groups of free groups, denoted Out(Fn). Recently, Algom-Kfir and Bestvina proposed an analog of Out(Fn) in the infinite-type setting: the mapping class group of a locally finite, infinite graph is the group of proper self homotopy equivalences, up to proper homotopy. These groups are not finitely generated, so classical tools of geometric group theory don't apply. Instead, we turn to descriptive set theoretic techniques of Rosendal to study their large-scale geometry. We will also give an overview of known properties towards understanding the algebraic and quasi-isometric rigidty of these groups. This is joint work with George Domat and Sanghoon Kwak.

Wednesday, February 7

Jenya Sapir (Binghampton University)

Title: Geodesics on high genus surfaces

Abstract: In joint work with Ben Dozier, we study the behavior of closed geodesics on hyperbolic surfaces of large genus. In her celebrated thesis, Mirzakhani showed that the number of simple closed geodesics of length at most L grows asymptotically like the polynomial L6g-6, where g is the genus. On the other hand, the number of all closed geodesics grows exponentially, like eL/L (Delsart, Huber, Selberg). For large L, therefore, most curves are non-simple. However, when L is small relative to the genus, g, this is no longer the case. In this talk, we will explore what "not-too-long" closed geodesics on large genus hyperbolic surfaces look like.

Wednesday, February 14

Sam Taylor (Temple University)

Title: Fixed points of pseudo-Anosov maps

Abstract: Let F be a pseudo-Anosov homeomorphism of a hyperbolic surface S. In this talk, we’ll describe joint work with Tarik Aougab and Dave Futer that predicts the number of fixed points of F, up to constants that depend only on the surface S. If F satisfies a mild condition called “strongly irreducible,” then the logarithm of the number of fixed points of F is coarsely equal to its translation length on the Teichmuller space of S. Without this condition, there is still a coarse formula involving subsurface projections of F’s invariant laminations.

Wednesday, February 21

Carolyn Abbott (Brandeis University)

Title: Morse boundaries of CAT(0) cubical groups

Abstract: The visual boundary of a hyperbolic space is a quasi-isometry invariant that has proven to be a very useful tool in geometric group theory. When one considers CAT(0) spaces, however, the situation is more complicated, because the visual boundary is not a quasi-isometry invariant. Instead, one can consider a natural subspace of the visual boundary, called the (sublinearly) Morse boundary. In this talk, I will describe a new topology on this boundary and use it to show that the Morse boundary with the restriction of the visual topology is a quasi-isometry invariant in the case of (nice) CAT(0) cube complexes. This result is in contrast to Cashen’s result that the Morse boundary with the visual topology is not a quasi-isometry invariant of CAT(0) spaces in general. This is joint work with Merlin Incerti-Medici.

Wednesday, February 28

Gil Goffer (UCSD)

Title: Small cancellation methods in probabilistic group laws

Abstract: In various cases, a law (that is, a quantifiers free formula) that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law [x,y]=1 holds with probability larger than 5/8, must be abelian. In this talk, I will give an overview of probabilistic laws (namely, laws that hold with high probability) on finitely generated groups, and illustrate how small cancellation methods and partial Burnside groups can reveal unexpected phenomena. For instance, I will present a partial Burnside group in which the law x^p=1 holds with probability 1, but yet the group does not satisfy this law, or any other law, globally, and a partial burnside group in which the probability that the law x^p=1 is satisfied oscillates between 0 and 1. This work, joint with Be’eri Greenfeld, is answering questions of Amir, Blachar, Gerasimova, and Kozma.

Wednesday, March 6

Kate Rybak (Vanderbilt)

Title: Frattini subgroups of hyperbolic-like groups.

Abstract: The Frattini subgroup Φ(G) of a group G is the intersection of all maximal subgroups of G; if G has no maximal subgroups, Φ(G) = G by definition. Frattini subgroups of groups with “hyperbolic-like” geometry are often small in a suitable sense. Generalizing several known results, we prove that for any countable group G admitting a general type action on a hyperbolic space S, the induced action of the Frattini subgroup Φ(G) on S has bounded orbits; in particular, Φ(G) has infinite index in G. In contrast, we show that the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups. The talk is based on a joint work with Gil Goffer and Denis Osin.

Wednesday, March 20

Tam Cheetham-West (Yale)

Title: Finite quotients of four-punctured sphere bundle groups

Abstract: The finite quotients of the fundamental group of a 3-manifold are the deck groups of its finite regular covers. We often pass to these finite-sheeted covers for different reasons, and these deck groups are organized into a topological group called the profinite completion of a 3-manifold group. In this talk, we will discuss how to leverage certain properties of the mapping class group of the four-punctured sphere to study the profinite completions of the fundamental groups of fibered hyperbolic four-punctured sphere bundles over the circle.

Wednesday, March 27

Tianyi Zheng (UCSD)

Title: Furstenberg entropy spectrum of stationary actions

Abstract: In this talk, we will discuss some aspects of the question: given a group, what is the range of the Furstenberg entropy of ergodic stationary actions of it? For the special linear group and its lattices, constraints on this spectrum come from Nevo-Zimmer structure theorems; and entropy values can be realized based on Poisson bundles over stationary random subgroups.

Wednesday, April 3

Talia Fernos (University of North Carolina, Greensboro)

Title: The Semi-Simple Theory of Acylindricity in Higher rank

Abstract: Acylindricity may be viewed as a generalization of being a uniform lattice in a locally compact second countable group. The theory of acylindrical actions on hyperbolic spaces has seen an explosion in recent years. Trees are of course examples of hyperbolic spaces, and by considering products, we start to see new and interesting behaviors that are not present in rank-1, such as the simple Burger-Mozes-Wise lattices, or Bestvina-Brady kernels. In a joint worth with S. Balasubramanya we introduce a new class of nonpositively curved groups. Viewing the theory of S-arithmetic semi-simple lattices as inspiration, we extend the theory of acylindricity to higher rank and consider finite products of delta-hyperbolic spaces. The category is closed under products, subgroups, and finite index over-groups. Weakening acylindricity to AU-acylindricity (i.e. acylindricity of Ambiguous Uniformity) the theory captures all S-arithmetic semi-simple lattices with rank-1 factors, acylindrically hyperbolic groups, HHGs, and many others. In this talk, we will discuss the Tits alternative and the associated outer-automorphism groups, and where to look for lattice envelopes according to the work of Bader-Furman-Sauer.

Wednesday, April 10

Lei Chen (University of Maryland)

Title: Mapping class groups of circle bundles over a surface

Abstract: In this talk, we study the algebraic structure of mapping class group Mod(M) of 3-manifolds M that fiber as a circle bundle over a surface. We prove an exact sequence, relate this to the Birman exact sequence, and determine when this sequence splits. We will also discuss the Nielsen realization problem for such manifolds and give a partial answer. This is joint work with Bena Tshishiku and Alina Beaini.

Wednesday, April 17

Jason Behrstock (CUNY)

Title: Quasiflats and quasi-isometric rigidity in hierarchically hyperbolic spaces

Abstract: Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, and others. In this talk, I'll provide an introduction to studying groups and spaces from this point of view. This discussion will center around work in which we classify quasiflats in these spaces, thereby resolving a number of well-known questions and conjectures. In particular, I will discuss how this study of quasiflats can be used to prove quasi-isometric rigidity for certain families of groups. This is joint work with Mark Hagen and Alessandro Sisto.

Wednesday, April 24

Ilya Kapovich (CUNY)

Title: On two-generator subgroups of mapping torus groups

Abstract: Motivated by the results of Jaco and Shalen about 3-manifold groups, we prove that if F is a free group (of possibly infinite rank), φ: F → F is an injective endomorphism of F and G = ⟨ F, t | txt−1 = φ(x), x ∈ F⟩ is the mapping torus group of φ, then every two-generator subgroup of G is either free or a “sub-mapping torus.” For a fully irreducible automorphism φ of a finite rank free group F this result implies that every two-generator subgroup of the free-by-cyclic group G is either free, free abelian, a Klein bottle group or a subgroup of finite index in G; and if φ ∈ Out(F) is fully irreducible and atoroidal then every two-generator subgroup of G is either free or of finite index in G. This talk is based on joint work with Naomi Andrew, Edgar A. Bering IV, and Stefano Vidussi.