Fall 2017 – GGTT Seminar

Seminar meets Tuesdays at 4:30 in BP 101 in the Tufts Math Department. You can subscribe to the GGTT mailing list here. In Fall 2017, the seminar organizers are Robert Kropholler and Genevieve Walsh.

Date	Name	Title
Sept 19	Kim Ruane	Boundaries of CAT(0) IFP Groups Abstract: I will discuss joint work with C. Hruska on boundaries of CAT(0) groups with the Isolated Flats Property. Such groups are known to be hyperbolic relative to the family of virtually abelian subgroups of rank greater than or equal to 2. In much older joint work, Mihalik and Ruane prove that if G is a one-ended CAT(0) group that has a “geometric splitting”, then any CAT(0) space on which G acts geometrically must have non-locally connected boundary. In the joint work with Hruska we prove a full converse to the theorem of Mihalik and Ruane in the setting of CAT(0) IFP groups – namely, that if G is a one-ended CAT(0) with IFP, then G has non-locally connected boundary if and only if G has admits a geometric splitting in the sense of Mihalik-Ruane. In the talk I will give examples of one-ended CAT(0) IFP groups with locally connected boundary and those with non-locally connected boundary. I will outline the proof of our theorem and discuss the main tools used in the proof..
Sept 26	Corey Bregman	Abelian-by-Surface groups and Kaehler Manifolds Abstract: A question of Serre asks which finitely generated groups arise as fundamental groups of compact Kaehler manifolds. We will begin with a survey of known results in this area, aimed at geometric group theorists, and highlight some connections to low-dimensional topology. We will then study extensions of abelian groups by hyperbolic surface groups. Topologically, these groups are realized as fundamental groups of surface bundles over tori. We show that if any such an extension is Kaehler, then it is virtually a product. This is joint work with Letao Zhang.
Oct 3	Sang-Hyun Kim	Free Products in $\mathrm{Diff}(S^1)$ Abstract: We prove that the class of subgroups of $\mathrm{Diff}^r(S^1)$ is not closed under taking free products for each $2\leq r \leq \infty$ . More specifically, $(F_2 \times Z) \ast Z$ does not embed into $\mathrm{Diff}^r(S^1)$ . We then complete the classification of RAAGs embeddable in $\mathrm{Diff}^r(S^1)$ , answering a question (of Kharlamov) in a paper of M. Kapovich. This is a joint work with Thomas Koberda.
Oct 10	Emily Stark	The visual boundary of hyperbolic free-by-cyclic groups Abstract: Given an automorphism of the free group, we consider the mapping torus defined with respect to the automorphism. If the automorphism is atoroidal, then the resulting free-by-cyclic group is hyperbolic by work of Brinkmann. In addition, if the automorphism is fully irreducible, then work of Kapovich–Kleiner proves the boundary of the group is homeomorphic to the Menger curve. However, their proof is very general and gives no tools to further study the boundary and large-scale geometry of these groups. In this talk, I will explain how to construct explicit embeddings of non-planar graphs into the boundary of these groups whenever the group is hyperbolic. This is joint work with Yael Algom-Kfir and Arnaud Hilion.
Oct 17	Phil Wesolek	Tree almost automorphism groups: elements and subgroups Abstract: (Joint work with A. Le Boudec) We begin by giving a detailed overview of the tree almost automorphism groups, which are rather exotic locally compact groups. In particular, we describe their relationship to the Higman-Thompson groups. We then discuss several structure results for elements and subgroups. For example, each almost automorphism comes in exactly one of two types, analogous to the elliptic or hyperbolic dichotomy for the classical case of tree automorphisms. As applications, we recover a result for Thompson’s group V as well as a new observation about the Röver group.
Oct 24	Ignat Soroko	Uncountably many quasi-isometry classes of groups of type FP Abstract: An interplay between algebra and topology goes in many ways. Given a space $X$ , we can study its homology and homotopy groups. In the other direction, given a group $G$ , we can form its Eilenberg-Maclane space $K(G,1)$ . It is natural to wish that it is ‘small’ in some sense. If $K(G,1)$ space has $n$ -skeleton with finitely many cells, then $G$ is said to have type $F_n$ . Such groups act naturally on the cellular chain complex of the universal cover for $K(G,1)$ , which has finitely generated free modules in all dimensions up to n. On the other hand, if the group ring $\mathbb{Z}G$ has a projective resolution $(P_i)$ of length $n$ where each module $P_i$ is finitely generated, then $G$ is said to have type $FP_n$ . There have been many intriguing questions on whether classes $F_n$ and $FP_n$ are different, and some of them are still open. Bestvina and Brady gave first examples of groups of type $FP_2$ which are not finitely presentable (i.e. not of type $F_2$ ). In his recent paper, Ian Leary has produced uncountably many of such groups. Using Bowditch’s concept of taut loops in Cayley graphs, we show that Ian Leary’s groups actually form uncountably many classes up to quasi-isometry. This is a joint work with Robert Kropholler and Ian Leary.
Oct 31	Arman Darbinyan	The word and conjugacy problems in lacunary hyperbolic groups Abstract: The class of lacunary hyperbolic (LHG) groups consists exactly of the finitely generated groups with at least one asymptotic cone being a real tree. They can be thought of as direct limits of hyperbolic groups with some `mild’ extra conditions. In my talk, I will concentrate on the word and conjugacy problems in LHG and explain how the study of these decision problems within LHG helps us to obtain new results in the well-explored area of word and conjugacy problems for finitely generated groups.
Nov 7	Kathryn Lindsey	Entropy, Galois conjugates, and postcritically finite maps Abstract: Generalized $\beta$ -transformations are self-maps of the unit interval that are formed by taking the transformation $x \mapsto \beta x \mod 1$ , where $\beta> 1$ , and replacing some of the branches with branches of constant negative slope. Such a map is called postcritically finite if the orbit of the point 1 is a finite set. The exponential of the topological entropy of a postcritically finite generalized $\beta$ -transformation is an algebraic integer. Building on work of Thurston, Tiozzo, Thompson and others, we investigate the question: what does the closure of the set of all Galois conjugates of all postcritically finite generalized $\beta$ -transformations look like? I will present some characterizations of this set. This talk is based on joint work with H. Bray, D. Davis and C. Wu.
Nov 14	Nick Salter	Subgroups of the mapping class group via algebraic geometry Abstract: This is a story about how an object beloved to geometric group theorists (the mapping class group) connects with the wider world of mathematics (particularly, algebraic geometry). The story begins with the basic observation that a “generic” polynomial in two variables determines a Riemann surface, and so loops of such polynomials determine mapping classes. The motivating question here is very simple: which mapping classes arise in this way? Perhaps surprisingly, the answer turns out to be “either none at all (for trivial reasons) or else virtually all of them”. No familiarity with algebraic geometry will be assumed.
Nov 21	Joseph Maher	Random subgroups of acylindrical groups Abstract: We consider a simple model of random subgroups of an acylindrical group, and show that generically they are hyperbolically embedded and satisfy a small cancellation condition. This is joint work with Alessandro Sisto.
Nov 28	Teddy Einstein	Hierarchies of Non-Positively Curved Cube Complexes Abstract: A non-positively curved (NPC) cube complex is a combinatorial complex constructed by gluing Euclidean cubes along faces in a way that satisfies a combinatorial local non-positive curvature condition. A hierarchy is an inductive method of decomposing the fundamental group of a cube complex. Cube complexes and hierarchies of cube complexes have been studied extensively by Wise and feature prominently in Agol’s proof of the Virtual Haken Conjecture for hyperbolic 3-manifolds. In this talk, I will give an overview of the geometry of cube complexes, explain how to construct a hierarchy for a NPC cube complex, and discuss applications of cube complex hierarchies to hyperbolic and relatively hyperbolic groups.
Dec 5	Meng-Che “Turbo” Ho	The word problem of a group as formal languages Abstract:The word problem of a group $G = <S>$ can be defined as the set of formal words in $S^*$ that represent the identity in $G$ . When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anisimov showed that a group is finite if and only if its word problems is regular, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is context-free. Recently, Salvati showed that the word problem of $\mathbb{Z}^2$ is multiple context-free. Afterwards, Kropholler and Spriano show that the class of groups with multiple context-free word problems is closed under amalgamated free products over finite groups. Gilman, Kropholler, and Schleimer showed that most nilpotent groups, RAAGs, and hyperbolic three-manifold groups do not have multiple context-free word problem. In this talk, we will discuss a generalization of Salvati’s result to show that $\mathbb{Z}^n$ has multiple context-free word problem.
Dec 12