Vanderbilt University

Fall 2020

Links to seminar schedule for previous years:

2010/11,
2011/12,
2012/13,
2014/15,
Spring 2016,
Fall 2016,
Spring 2017,
2017/18,
2018/19
2019/20

**Organizers:** Spencer Dowdall & Denis Osin

**Time:** Wednesdays, 5:10–6:00pm US Central Time

**Format:** Due to COVID the seminar will meet **Online via Zoom Meeting**.

Please email one of the organizers for the Zoom Meeting invitation and join link.

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

Denis Osin (Vanderbilt University)

Title: A topological zero-one law and elementary equivalence of finitely generated groups

Abstract: Let \(\mathcal G\) denote the space of finitely generated marked groups. We will discuss equivalent characterizations of closed subspaces \(\mathcal S\subseteq \mathcal G\) satisfying the following zero-one law: for any sentence \(\sigma\) in the infinitary logic \(\mathcal L_{\omega_1, \omega}\), the set of all models of \(\sigma\) in \(\mathcal S\) is either meager or comeager. In particular, I will prove that the zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. As an application, I will show that generic torsion-free lacunary hyperbolic groups are elementarily equivalent; the same claim holds for lacunary hyperbolic groups without non-trivial finite normal subgroups. We will also discuss some open problems.

Rose Morris-Wright (UCLA)

Title: An analog to the curve complex for FC type Artin groups

Abstract: Parabolic subgroups are the fundamental building blocks of Artin groups. These subgroups are isomorphic copies of smaller Artin groups nested inside a given Artin group. In this talk, I will discuss questions surrounding how parabolic subgroups sit inside Artin groups and how they interact with each other. I will show that, in an FC type Artin group, the intersection of two finite type parabolic subgroups is a parabolic subgroup. I will also discuss how parabolic subgroups might be used to construct a simplicial complex for Artin groups similar to the curve complex for mapping class groups. This talk will focus on using geometric techniques to generalize results known for finite type Artin groups to Artin groups of FC type.

Frank Wagner (Vanderbilt)

Title: Torsion Subgroups of Groups with Quadratic Dehn Function

Abstract: The Dehn function of a finitely presented group, first introduced by Gromov, is a useful invariant that is closely related to the solvability of the group’s word problem. It is well-known that a finitely presented group is word hyperbolic if and only if it has sub-quadratic (and thus linear) Dehn function. A result of Ghys and de la Harpe states that no word hyperbolic group can have a (finitely generated) infinite torsion subgroup. We show that this property does not carry over to any class of groups of larger Dehn function. In particular, for every m>1 and n sufficiently large (and either odd or divisible by 2^9), there exists a quasi-isometric embedding of the infinite free Burnside group B(m,n) into a finitely presented group with quadratic Dehn function.

Elizabeth Field (University of Utah)

Title: Trees, dendrites, and the Cannon-Thurston map

Abstract: When \(1\to H\to G\to Q\to 1\) is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from \(H\) to \(G\) extends continuously to a map between the Gromov boundaries of \(H\) and \(G\). This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point \(z\) in the Gromov boundary of \(Q\) an "ending lamination" on \(H\) which consists of pairs of distinct points in the boundary of \(H\). We prove that for each such \(z\), the quotient of the Gromov boundary of \(H\) by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where \(H\) is a free group and \(Q\) is a convex cocompact purely atoroidal subgroup of \(\mathrm{Out}(F_N)\), one can identify the resultant quotient space with a certain \(\mathbb{R}\)-tree in the boundary of Culler-Vogtmann's Outer space.

Ruth Charney (Brandeis)

Title: Outer Space for Right-Angled Artin Groups

Abstract: Right-angled Artin groups (RAAGs) span a range of groups from free groups to free abelian groups. Thus, their (outer) automorphism groups range from Out(F_n) to GL(n,Z). Automorphism groups of RAAGs have been well-studied over the past decade from a purely algebraic viewpoint. To allow for a more geometric approach, one needs to construct a contractible space with a proper action of the group. I will talk about joint work with Bregman and Vogtmann in which we construct such a space, namely, an analogue of Culler-Vogtmann’s Outer Space for arbitrary RAAGs.

The Zoom recording of this talk can be (temporarily) viewed here: https://vanderbilt.zoom.us/rec/share/GY-EjWVI7446x3GCFMKfiM8vulK_6reVJB6ufuo3VKhL3zNlknW1d7KlQ4Xp3eT6.CMrgApSofDBXZvdS.

Alex Margolis (Vanderbilt)

Title: Topological completions of quasi-actions and discretisable spaces

Abstract: A fundamental problem in geometric group theory is the study of quasi-actions. We introduce and investigate discretisable spaces: spaces for which every cobounded quasi-action can be quasi-conjugated to an isometric action on a locally finite graph. Work of Mosher-Sageev-Whyte shows that non-abelian free groups are discretisable, but the property holds much more generally. For instance, every non-elementary hyperbolic group that is not virtually isomorphic to a cocompact lattice in rank one Lie group is discretisable.

Along the way, we study the coarse geometry of groups containing almost normal/commensurated subgroups, and we introduce the concept of the topological completion of a quasi-action. The topological completion is a locally compact group, well-defined up to a compact normal subgroup, reflecting the geometry of the quasi-action. We give several applications of the tools we develop. For instance we show that any finitely generated group quasi-isometric to a Z-by-hyperbolic group is also Z-by-hyperbolic, and prove quasi-isometric rigidity for a large class of right-angled Artin groups.

The Zoom recording of this talk can be (temporarily) viewed here: https://vanderbilt.zoom.us/rec/share/0LGQ4ctOnQp1htjCkla-N20gIuIFk8eNUhqe4lfswUf-HmOHO-T-Vo9onalOsKU9.mCQcn906oQnHbxyo.

Alexander J. Rasmussen (University of Utah)

Title: Actions of certain solvable groups on hyperbolic metric spaces

Abstract: In this talk I will discuss the classification of the cobounded hyperbolic actions of solvable Baumslag-Solitar groups and lamplighter groups. These groups have actions on trees which have convenient descriptions using ring theory. I will then discuss current work on generalizing this ring theoretic machinery, and possible applications to certain abelian-by-cyclic groups. This is joint work with Carolyn Abbott and Sahana Balasubramanya.

The Zoom recording of this talk can be (temporarily) viewed here: https://vanderbilt.zoom.us/rec/share/D3XbcOB5tjLBs5tVPXXNm4_CaTGLsgc9bnimwORg67J5CfkJkqVoFgyTbg5Eb1U8._ByxEGdSD7XpZM2-.

Rémi Coulon (CNRS Rennes)

Title: Automorphisms with exotic growth

Abstract: Let \(G\) be a finitely generated group. Given an (outer) automorphism \(\phi\) of \(G\), one can study its properties by considering the dynamics induced by the action of \(\phi\) on the set of conjugacy classes of \(G\). A classical problem is to understand how the length of a conjugacy class grows under the iterations of \(\phi\). For example, if \(G\) is a surface group and \(\phi\) the automorphism induced by a homeomorphism \(f\), this amounts to studying the length of closed geodesics under the iterations of \(f\). For many groups (e.g, free groups, free abelian groups, surface groups, etc) one observes a strong dichotomy : the length of any conjugacy class grows either polynomially or at least exponentially. In this talk, we will explain how to build examples of outer automorphisms of finitely generated groups for which this dichotomy fails. We will see that any reasonable type of growth can actually be achieved.

Wenhao Wang (Vanderbilt)

Title: Dehn Functions of Finitely Presented Metabelian Groups

Abstract: The Dehn function was introduced by computer scientists Madlener and Otto to describe the complexity of the word problem of a group, and also by Gromov as a geometric invariant of finitely presented groups. In this talk, I will show that the upper bound of the Dehn function of finitely presented metabelian group \(G\) is \(2^{n^{2k}}\), where \(k\) is the torsion-free rank of the abelianization \(G_{ab}\), answering the question that if the Dehn functions of metabelian groups are uniformly bounded. I will also talk about the relative Dehn function of finitely generated metabelian group and its relation to the Dehn function.