We will talk about the widely unknown classification of general surfaces due to Brown and Messer. Then we will discuss how the classification was used to get general results about the mapping class groups of orientable surfaces. In particular, we classified the automatically continuous pure mapping class groups over all orientable surfaces.

Surface Houghton groups are a generalization of Houghton groups to the surface setting. They are defined as groups of asymptotically rigid mapping classes of an infinite-type surface. We will give commensurability and isomorphism classication results for this class of groups. Some time will be spent motivating these groups as "medium" mapping class groups that live somewhere between the world of mapping class groups of finite-type surfaces and those of infinite-type surfaces. This is joint work with Javier Aramayona and Christopher Leininger.

We introduce a 1-complex $\mathrm{Wh}(\Gamma)$Wh(Γ) associated with a finite regular cover $\Gamma$Γ of the rose which is connected if and only if the fundamental group of the associated cover is generated by elements in a proper free factor of the free group. When the associated cover represents a characteristic subgroup of the free group, the complex admits an action of $\mathrm{Out}(F_n)$Out(Fn​) by isometries. We then explore the coarse geometry of $\mathrm{Wh}(\Gamma)$Wh(Γ). Every component of $\mathrm{Wh}(\Gamma)$Wh(Γ) has infinite diameter, and the complex $\mathrm{Wh}(\mathbf{R}_n)$Wh(Rn​) associated with the rose is nonhyperbolic. As corollaries, we obtain that the Cayley graph of the free group with the infinite generating set consisting of all primitive elements has infinite diameter and is nonhyperbolic.

In this talk, I will define and motivate the use of horofunction boundaries to study groups. I will discuss some examples which demonstrate interesting properties of the horofunction boundary and share new results about the horofunction boundaries of homogeneous groups.

Shephard groups are common generalizations of Coxeter groups, Artin groups, graph products of cyclic groups, and (certain) complex reflection groups. We extend a well-known result that Coxeter groups are CAT(0) to a class of Shephard groups that have “enough” finite parabolic subgroups. We also show that in this setting, if the underlying Coxeter diagram is type FC, then the Shephard group is (cocompactly) cubulated. Our method of proof combines the works of Charney-Davis on the Deligne complex for an Artin group and of Coxeter on the classification and properties of the regular complex polytopes.

The interest in the geometry of group extensions started with the geometrization theorem of Thurston for compact irreducible atoroidal 3-manifolds. We will talk about the geometry of group extensions and the motivations behind such studies in the cases of closed surface groups and free groups. More specifically, we will talk about the most general case so far and, we will give necessary and sufficient conditions for a free extension of a (non-Abelian) free group given by a subgroup of the outer automorphism group of the free group (Out(F_n)) to be hyperbolic and relatively hyperbolic. Joint work with Pritam Ghosh.

The central limit theorem of random walks answers the question "how quickly does the random walk drift away from the origin". Historically, it has been proven (under some assumptions) for free groups, hyperbolic groups, and various generalizations of hyperbolic groups. We proved this for one generalization of hyperbolic groups: groups containing superlinear divergent quasi-geodesics. The advantage of this setting compared to previous versions of CLT is that it is invariant under quasi-isometry.

In this talk, I will delve into the superlinear divergence property, as well as its geometric consequences that led to the theory of random walks on groups containing superlinear divergent quasi-geodesics. This talk is based on joint work with Kunal Chawla, Inhyeok Choi, and Kasra Rafi.

In 1911, Dehn proposed three decision problems for finitely presented groups: word problem, conjugacy problem, and isomorphism problem. These problems have been central to both group theory and logic, and were each proven to be undecidable in the 50s. There is much current research studying the decidability of these problems in classes of groups.

Although these problems are classically studied as decision problems, each of them is naturally an equivalence relation. In this talk, we study them as equivalence relations and compare them using computable reductions. This leads to a more refined measure of their complexity and brings new results and questions.

The mapping class group Mod(S) of a surface S acts on the homology H_1(S), yielding the well-studied symplectic representation Mod(S) → Sp(2g,Z). In this talk, we discuss an equivariant refinement of the symplectic representation. Namely, given a finite group G acting on S, the symplectic representation restricts to a map from the centralizer of G in Mod(S) to the centralizer of G in Sp(2g,Z). The image of this restriction has been studied by many authors and is generally difficult to understand. We discuss our result that the image of this restriction is arithmetic when S has genus at most 3.

A finitely generated group can be thought of as a metric space when equipped with the word metric with respect to a finite generating set. This metric space is well-defined up to quasi-isometry. A major program in geometric group theory, initiated by Gromov, is determining to what extent the coarse geometry of a group determines its algebra. In this talk, we investigate when normal and commensurated subgroups, and their associated quotient groups and spaces, are preserved by quasi-isometries.

The topology of the Bowditch boundary of a relatively hyperbolic group pair gives information about relative splittings of the group. It is therefore interesting to ask if there is generic behavior of this boundary. In this talk I plan to describe previously known results about the Bowditch boundary of a free group with cyclic peripheral structure, and discuss why there is no generic case when the peripheral structure is produced randomly.

The Torelli group of a surface is the kernel of the action of the mapping class group on the first homology of the surface. It is a longstanding open problem to determine whether or not the Torelli group is finitely presented for closed oriented surfaces of genus at least 3. We will discuss some recent work of the author that rules out the simplest obstruction to the Torelli group being finitely presented. In particular, we will show that the second rational homology of the Torelli group is finite dimensional for all surfaces of sufficiently large genus.

The notion of a girth was first introduced by S. Schleimer in 2003. Later, a substantial amount of work on the girth of finitely generated groups was done by A. Akhmedov, where he introduced the so-called Girth Alternative and proved it for certain classes of groups, e.g. hyperbolic, linear, one-relator, $PL_+(I)$PL+​(I) etc. Girth Alternative is similar to the well-known Tits Alternative in spirit, therefore it is natural to study it for classes of groups for which Titis Alternative has been investigated. In this talk, we will explore the girth of HNN extensions of finitely generated groups in its broadest sense by considering cases where the underlying subgroups are either full or proper subgroups. We will present a sub-class for which Girth Alternative holds. We will also produce counterexamples to show that beyond our class, the alternative fails in general. Recently, we extended one of the main results proving the Girth Alternative for HNN extensions of word hyperbolic groups (instead of HNN extensions of free groups). The talk will be based on joint work with Azer Akhmedov.

Quotients of free products are natural combinations of groups that have been exploited to study embedding problems. These groups have seen a resurgence of attention from a more geometric point of view following celebrated work of Haglund--Wise and Agol. I will discuss a geometric model for studying quotients of free products. We will use this model to adapt ideas from Gromov's density model to this new class of quotients, their actions on CAT(0) cube complexes, and combination theorems for residual finiteness. Results discussed will be based on ongoing work with Einstein, Krishna MS, Montee, and Steenbock.

A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by Borel 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, actions of mapping class groups on arc graphs and curve graphs, 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.

A group is "coherent" if every finitely generated subgroup is finitely presented. In a certain sense, coherence is a low-dimensional phenomenon. For instance, 3-manifold groups and one-relator groups are coherent. In this talk we consider Coxeter groups defined by a graph that is a flag triangulation of a surface of genus g. For each g>0, we construct a Coxeter group that is right-angled, hyperbolic, and incoherent. In these examples the witness to incoherence is always the fiber in a virtual algebraic fibration. This provides positive evidence towards a variation on Singer's Conjecture for right-angled Coxeter groups proposed by Davis-Okun. This is joint work with G. Walsh.

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves in the surface and whose edges connect disjoint curves. Following a rich history of hyperbolicity in various graphs based on surfaces, the fine curve was shown to be hyperbolic by Bowden–Hensel–Webb. Given how well-studied the curve graph and the case of “up to isotopy” is, we ask: what about the part of the fine curve graph not captured by isotopy classes? In this talk, we introduce the result that the subgraph of the fine curve graph spanned by curves in a single isotopy class is not hyperbolic; indeed, it contains a flat of EVERY dimension. Joint work with Ryan Dickmann.

We consider the collection of parabolically geometrically finite (PGF) subgroups of mapping class groups, which were defined by Dowdall-Durham-Leininger-Sisto. These are generalizations of convex cocompact groups, and the class of PGF groups contains all finitely generated Veech groups as well as certain free products of multitwist groups. We will see some basic motivations and properties of these groups, as well as discuss a combination theorem for PGF groups generalizing the combination theorem of Leininger-Reid for Veech groups. This allows one to build many more examples of PGF groups, including Leininger-Reid surface groups.

In this talk we will discuss how the minimal area of almost Fuchsian subgroups (more precisely, their asymptotic growth) of a Kleinian group detects the hyperbolic metric under pinched curvature conditions. This is based on upcoming joint work with Ruojing Jiang.

In 1982, Leighton proved that any two finite graphs with a common cover admit a finite sheeted common cover. In this talk, I will introduce the combinatorial model $X_{m,n}$Xm,n​ for the Baumslag-Solitar group BS(m,n), and classify for which pairs of integers (m,n) Leighton's theorem can be extended to the orbit space of covering actions on $X_{m,n}$Xm,n​.

Let S be a compact orientable connected surface with negative Euler characteristic. Two closed curves on S are of the same type if their corresponding free homotopy classes differ by a mapping class of S. Given two closed curves on S, we propose an algorithm to detect whether they are of the same type or not. This is joint work with Juan Souto.