Topology & Group Theory Seminar

Vanderbilt University

Spring 2017

Organizers: Gili Golan, Andrew Sale

Wednesdays, 4:10pm in SC 1308

** Wednesday, January 18, 2017 **

Craig Guilbault (University of Wisconsin-Milwaukee)

Title: Infinite boundary connected sums and aspherical manifolds.

Abstract: A boundary connected sum \(Q_{1}\lozenge Q_{2}\) of n-manifolds is obtained by gluing \(Q_{1}\) to \(Q_{2}\) along \((n-1)\)-balls contained in the respective boundaries. It is an elementary fact of manifold topology that, under mild conditions, this gives a well-defined operation that is both commutative and associative. In particular (under appropriate conditions) the boundary connected sum \(\lozenge_{i=1}^{k}Q_{i}\) of a finite collection of n-manifolds is topologically well-defined. This observation fails spectacularly when we attempt to generalize it to countable collections. In this talk I will discuss a reasonable substitute for a well-definedness theorem for infinite boundary connected sums. As one application we will exhibit some closed aspherical manifolds with exotic, i.e., not homeomorphic to $R^n$, universal covers that are unlike those found in the classical examples produced by Davis and Davis-Januszkiewicz. All of this work is joint with Ric Ancel and Pete Sparks.

Spencer Dowdall (Vanderbilt)

Title: Hyperbolicity in Outer space with applications to free group extensions

Abstract: The "Outer space" of the rank n free group F_n is a contractible metric space on which the Outer automorphism group Out(F_n) acts properly discontinuously. It was introduced by Culler and Vogtmann in 1986 and is now an important tool for the topological and geometric study of Out(F_n).

This talk will focus on the geometry of Outer space and implications for free group extensions. The first aspects of hyperbolicity in Outer space were discovered by Algom-Kfir, who showed that axes of fully irreducible automorphisms are strongly contracting. In this talk I will present a characterization of this strongly contracting property in terms of the geodesic's projection to the free factor complex. This characterization allows one to exploit the hyperbolicity of Outer space to study many geometric aspects of free group extensions. Results here include a flexible means of producing hyperbolic free group extensions, qualitative statements regarding their structure and quasiconvex subgroups, and quantitative results about their Cannon-Thurston maps. Mostly joint with Sam Taylor, and some joint with Ilya Kapovich and Sam Taylor.

Dan Margalit (Georgia Tech)

Title: Models for Mapping Class Groups

Abstract: A celebrated theorem of Nikolai Ivanov states that the automorphism group of the mapping class group is again the mapping class group. The key ingredient is his theorem that the automorphism group of the complex of curves is the mapping class group. After many similar results were proved, Ivanov made a metaconjecture that any “sufficiently rich object” associated to a surface should have automorphism group the mapping class group. In joint work with Tara Brendle, we show that the typical normal subgroup of the mapping class group has automorphism group the mapping class group. To do this, we show that a large family of complexes associated to a surface has automorphism group the mapping class group.

Grace Work (Vanderbilt)

Title: Constructing Transversals to Horocycle Flow

Abstract: Examining the distribution of the gaps between elements of a sequence can provide insight into how equidistributed these elements are. In the setting of translation surfaces, an important sequence is that of slopes of saddle connections. This distribution has been computed for specific examples, the square torus by Athreya and Cheung, and the double pentagon, by Athreya, Chaika and Lelievere. The strategy of proof involves reinterpreting the question in the setting of horocycle flow on the moduli space. Specifically, the gaps can be seen as return times under horocycle flow to a transversal. In joint work with Caglar Uyanik, we compute the distribution in the octagon and then provide a parametrization for the transversal for any lattice surface that depends on the parabolic elements of the Veech group. In the case of a generic surface, the situation becomes more complicated.

Mike Mihalik (Vanderbilt)

Title: Bounded Depth Ascending HNN Extensions and $\pi_1$-Semistability at Infinity

Abstract: Semistable fundamental group at $\infty$ for a finitely presented group $G$ is an asymptotic geometric condition for $G$ that allows one to unambiguously define the fundamental group at $\infty$ for $G$. A long standing open problem asks if all finitely presented groups have semistable fundamental group at $\infty$. If $G$ is an ascending HNN extension of a finitely presented group then indeed, $G$ has semistable fundamental group at $\infty$, but since the early 1980's it has been suggested that the finitely presented groups that are ascending HNN extensions of *finitely generated* groups may include a group with non-semistable fundamental group at $\infty$. Ascending HNN extensions naturally break into two classes, those with bounded depth and those with unbounded depth. Our main theorem shows that bounded depth finitely presented ascending HNN extensions of finitely generated groups have semistable fundamental group at $\infty$. Semistability is equivalent to two weaker asymptotic conditions on the group holding simultaneously. We show one of these conditions holds for all ascending HNN extensions, regardless of depth. We give a technique for constructing ascending HNN extensions with unbounded depth. This work focuses attention on a class of groups that may contain a group with non-semistable fundamental group at $\infty$.

Stephen G. Simpson (Vanderbilt)

Title: Symbolic dynamics: entropy = dimension = complexity

Abstract: Let $G$ be the additive group $\mathbb{Z}^d$ or the additive monoid $\mathbb{N}^d$ where $d$ is a positive integer. Let $X$ be a subshift over $G$, i.e., a nonempty, closed, shift-invariant subset of $A^G$ where $A$ is a finite alphabet. I prove that the topological entropy of $X$ is equal to the Hausdorff dimension of $X$. (The special case $G=\mathbb{N}$ of this result is due to H. Furstenberg in a 1967 paper.) I also obtain a sharp characterization of the Hausdorff dimension of $X$ in terms of the Kolmogorov complexity of finite pieces of the orbits of $X$. My paper is available as arXiv 1702.04394, and Nikita Moriakov and I are working to generalize my results to a larger class of groups.

Sahana Balasubramanya (Vanderbilt)

Title: Acylindrical structures on groups

Abstract: For every group G, we introduce the set of acylindrically hyperbolic structures on G, denoted AH(G). One can think of elements of AH(G) as cobounded acylindrical G-actions on hyperbolic spaces considered up to a natural equivalence. Acylindrically hyperbolic structures can be ordered in a natural way according to the amount of information they provide about the group G. We answer some basic questions about the poset structure of AH(G) and obtain several more advanced results about existence of maximal structures (or acylindrically hyperbolic accessibility), and rigidity phenomena similar to marked spectrum rigidity for hyperbolic manifolds.

Coauthors: Carolyn Abbott, Denis Osin

Oleg Bogopolski (Universität Düsseldorf)

Title: Generalized presentations of groups, in particular of $\mathrm{Aut}(F_{\omega})$.

Abstract: We introduce generalized presentations of groups. Roughly speaking, a generalized presentation of a group $G$ consists of a generalized free group $\mathcal{F}$ (which is a certain subgroup of a big free group ${\rm BF(\Lambda)}$) and of a subset $R$ of $\mathcal{F}$ such that $G$ is isomorphic to $\mathcal{F}/\overline{\langle\!\langle R\rangle\!\rangle}$, where $\overline{\langle\!\langle R\rangle\!\rangle}$ is the closure of $\langle\!\langle R\rangle\!\rangle$ with respect to an appropriate topology on $\mathcal{F}$.

We give a generalized presentation of $\mathrm{Aut}(F_{\omega})$, the automorphism group of the free group of infinite countable rank. This generalized presentation is countable, although the group itself is uncountable. We also give an account of known facts on $\mathrm{Aut}(F_{\omega})$ and formulate open problems.

This is a joint work with Wilhelm Singhof.

Arman Darbinyan (Vanderbilt)

Title: Word and Conjugacy Problems in Lacunary Hyperbolic Groups

Abstract: We study word and conjugacy problems in lacunary hyperbolic groups (LHG). In the talk we will discuss "if and only if" conditions for decidability of the word problem in LHG. Then we will discuss constructions of LHGs which have extreme behavior in terms of word and conjugacy problems. All these constructions are different manifestations of a group theoretical construction involving a version of small-cancellation theory.

In particular, we will also discuss how to construct Tarskii monsters and verbally complete groups as quotients of arbitrary non-elementary torsion-free hyperbolic groups with fast word and conjugacy problems.

Another application is that for any recursively enumerable subset $\mathcal{L} \subseteq \mathcal{A}^*$, where $\mathcal{A}^*$ is the set of words over arbitrarily chosen non-empty finite alphabet $\mathcal{A}$, there exists a lacunary hyperbolic group $G^{\mathcal{L}}$ such that the membership problem for $ \mathcal{L}$ can be in linear time reduced to the conjugacy problem in $G^{\mathcal{L}}$, and the inverse reduction can be done in 'almost' linear time. Moreover, for the mentioned group the word problem is decidable in 'almost' linear time.

We will also mention open questions which we have answered within this work.

Matthieu Jacquemet (Vanderbilt)

Title: Hyperbolic Coxeter groups with fundamental polyhedra of simple combinatorics

Abstract: We are interested in Coxeter groups which are isomorphic to subgroups of isometries of the (real) hyperbolic space and have fundamental domains of finite volume. Unlike their spherical and Euclidean cousins, these groups exist only in (relatively) low dimensions, and are far from being classified. This is particularly unfortunate since hyperbolic Coxeter groups are, in the known cases, related to small volume hyperbolic orbifolds.

One way to study this question is to consider it more geometrically/combinatorially, by studying hyperbolic Coxeter polyhedra. But as it turns out, even 'simple' families of such polyhedra remain cryptic. In this talk, I shall first outline some classic results about hyperbolic Coxeter polyhedra, and then provide the recently established classification of hyperbolic Coxeter n-cubes, resulting from a joint work with Steve Tschantz.

No seminar: department awards ceremony.