Title: Poset of acylindrical actions
Abstract: The class of groups that admit so-called acylindrical actions on hyperbolic spaces contains many interesting groups, such as mapping class groups, Out(F_n), right-angled Coxeter groups, and many more. Each such group will actually admit many acylindrical actions on different hyperbolic spaces, some of which are more useful than other for studying the group. In this talk, I will define an acylindrical action, and then describe how to put a partial order on the set of such actions to obtain a poset. I will then give some structural properties of this poset, and, in particular, will discuss for which groups a largest element of the poset exists.
Title: The geodesic flow on infinite type surfaces
Abstract: In this talk we first describe some of the known results on the geometry and topology of infinite (topological) type surfaces and then we investigate the relationship between Fenchel-Nielsen coordinates and when the geodesic flow on such a surface is ergodic. Ergodicity of the geodesic flow is equivalent to the surface being of so called parabolic type (the surface does not carry a Green’s function), and hence this problem is intimately connected to a version of the classical type problem in the study of Riemann surfaces. Specifically, we study so called tight flute surfaces - (possibly incomplete) hyperbolic surfaces constructed by linearly gluing infinitely many tight pairs of pants along their cuffs- and the relationship between their type and geometric structure. This is joint work with Hrant Hakobyan and Dragomir Saric.
Title: Convex cores of thick hyperbolic 3-manifolds with bounded rank
Abstract: I'll describe a geometric decomposition of the convex core of an epsilon-thick hyperbolic 3-manifold with bounded rank. As a consequence, every such convex core is (1,C)-quasi-isometric to a metric graph, for some uniform C.
Title: Geometrical finiteness and Veech subgroups of mapping class groups
Abstract: I will discuss work in progress with Dowdall, Leininger, and Sisto, in which we aim to develop a notion of geometrical finiteness for subgroups of mapping class groups. Motivated by the theory of convex cocompact subgroups, which are precisely those which determine hyperbolic surface group extensions, I will describe some hyperbolic properties of the surface group extensions coming from lattice Veech subgroups.
Title: Generic quotients of cubulated hyperbolic groups
Abstract: A group is called cubulated if it is the fundamental group of a compact, non-positively curved cube complex. Cubulated groups have many desirable properties, and have recently received much interest. We show that generic quotients of cubulated hyperbolic groups are again cubulated and hyperbolic. Here, “generic” means that the conclusion holds almost surely when long random relations are added at sufficiently low density, in the Gromov density model of random groups. This is joint work with Dani Wise.
Title: Corank of 3-manifold groups G with H_2(G)=0
Abstract: The corank of a group G, c(G), is the maximal r such that there is a surjective homomorphism from G to a non-abelian free group of rank r. We note that for any group G, c(G) is bounded above by b_1(G), the rank of the abelianization of G. For closed surface groups S, we have a further relationship between these two complexities, namely b_1(S) = 2 c(S). It was asked whether such a relationship exists for 3-manifold groups. In a previous paper, I showed that there were closed 3-manifold groups G with b_1(G) arbitrarily large but with c(G)=1. It was asked by Michael Freedman whether such a statement was known when the group was the group of a 3-dimensional homology handlebody. These groups are much more subtle and have properties that make them look like a free group so the question becomes much more difficult. In fact, all of the previous techniques used by the author fail. The complete answer to the question is still unknown. However, we show that there are groups G_m (for all m \geq 2) which are the fundamental group of a 3-dimensional handlebody (in particular, H_2(G_m)=0) and satisfy the following: b_1(G_m)= m and c(G_m)=f(m) where f(m)=m/2 for m even and f(m)=(m+1)/2 for m odd. This is joint work with Eamonn Tweedy.
Title: Skinning maps along thick rays
Abstract: For which countable groups G, does the moduli space
I'll discuss work in progress with K. Bromberg and Y. Minsky. We show that the diameter of the skinning map of an acylindrical 3-manifold along a thick ray in the Teichmueller space is bounded by constants depending only on the injectivity radius and genus of the boundary
Title: Rank 1 character varieties of finitely presented groups
Abstract: Let X(F,G) be the G-character variety of F where G is a rank 1 complex affine algebraic group and F is a finitely presentable discrete group. We describe an algorithm, which we implement in Mathematica, SageMath, and in Python, that takes a finite presentation for F and produces a finite presentation of the coordinate ring of X(F,G). This is joint work with Caleb Ashley, and Jean-Philippe Burelle.
Title: Exotic limit sets of Teichmuller geodesics in the HHS boundary
Abstract: We will present our result that a geodesic ray in Teichmuller space does not necessarily converge to a unique point in the hierarchically hyperbolic space (HHS) boundary of Teichmuller space, answering a question of Durham, Hagen, and Sisto. In fact, the limit set of a geodesic ray can be almost anything allowed by topology. This stands in contrast to the situation for Gromov hyperbolic spaces where each geodesic ray converges to a unique point in the HHS boundary (which happens to agree with the Gromov boundary).
Title: Counting problems in graph products
Abstract: Graph products of groups generalize both right-angled Artin and right-angled Coxeter groups. In this talk, I’ll discuss joint work with I. Gekhtman and G. Tiozzo which as a special case handles various counting problems in the Cayley graphs of such groups. For example, we show that when a right-angled Artin group is not a direct product, the proportion of rank 1 elements in the ball of radius n, with respect to the standard generators, goes to 1 as n approaches infinity.
Title: Coarse geometry of expanders from homogeneous spaces
Abstract: Subgroups of compact Lie groups give rise to expander graphs via a warped cone construction. We study the dependence of the coarse geometry of such expander graphs on the original subgroup and show they must exhibit a wide range of geometric behavior: The coarse geometry of the warped cone determines the subgroup up to conjugacy. As an application, we produce uncountably many non-quasi-isometric expanders. This is joint work with David Fisher and Thang Nguyen.