Topology & Group Theory Seminar

Vanderbilt University

2016/2017


Organizers: Mark Sapir (Fall), Gili Golan, Andrew Sale (Spring)

Wednesdays, 4:10pm in SC 1308

Spring 2017:


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.



Wednesday, January 25, 2017

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.



Wednesday, February 1, 2017

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.



Wednesday, February 8, 2017

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.



Wednesday, February 15, 2017

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 $\infty$.



Wednesday, February 22, 2017

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.




Wednesday, March 15, 2017

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



Wednesday, March 29, 2017

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.



Wednesday, April 5, 2017

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.



Wednesday, April 12, 2017

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.



Wednesday, April 19, 2017

No seminar: department awards ceremony.



Fall 2016:


Wednesday, August 31, 2016

Andrew Sale (Vanderbilt)

Title: When the outer automorphism groups of RAAGs are vast

Abstract:Right-angled Artin groups (RAAGs) are a class of groups that bridge the gap between free groups and free abelian groups. Thus, their outer automorphism groups give a way to build a bridge between GL(n,Z) and Out(Fn). We will investigate certain properties of these groups which could be described as a "vastness" property, and ask if it possible to build a boundary between those which are "vast" and those which are not. One such property is as follows: given a group G, we say G has all finite groups involved if for each finite group H there is a finite index subgroup of G which admits a map onto H. From the subgroup congruence property, it is known that the groups GL(n,Z) do not have every finite group involved for n>2. Meanwhile, the representations of Out(Fn) given by Grunewald and Lubotzky imply that these groups do have all finite groups involved. We will describe conditions on the defining graph of a RAAG that are necessary and sufficient to determine when it's outer automorphism group has this property. The same criterion also holds for other properties, such as SQ-universality, or having a finite index subgroup with infinite dimensional second bounded cohomology. This is joint work with V. Guirardel.

Wednesdays, September 7, 14, 2016

Denis Osin (Vanderbilt)

Title: Induced group actions on metric spaces.

Abstract: We will discuss the following natural extension problem for group actions: Given a group G, a subgroup H < G, and an action of H on a metric space S, when is it possible to extend it to an action of the whole group G on a possibly different metric space? When does such an extension preserve interesting properties of the original action of H? I will explain how to formalize this problem and will present a construction of the induced action of G which behaves well when G is hyperbolic relative to H or, more generally, H is hyperbolically embedded in G; in particular, the induced action solves the extension problem in these cases. This talk is based on a joint work with C. Abbott and D. Hume.

Wednesday, September 21, 2016

Gili Golan (Vanderbilt)

Title: The generation problem in Thompson group F

Abstract: We show that the generation problem in Thompson group F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F generates the whole F. The algorithm makes use of the Stallings 2-core of subgroups of F, which can be defined in an analogue way to the Stallings core of subgroups of a free group. An application of the algorithm shows that F is a cyclic extension of a group K which has a maximal elementary-amenable subgroup B.

Wednesday, October 5, 2016

John Ratcliffe (Vanderbilt)

Title: A Bieberbach theorem for crystallographic group extensions.

Abstract: Joint work with Steven Tschantz. We will talk about our relative Bieberbach theorem: For each dimension n there are only finitely many isomorphism classes of pairs of groups (Γ,N) such that Γ is an n-dimensional crystallographic group and N is a normal subgroup of Γ such that Γ/N is a crystallographic group. This result is equivalent to the statement that for each dimension n there are only finitely many affine equivalence classes of geometric orbifold fibrations of compact, connected, flat n-orbifolds.

Wednesday, October 26, 2016

Mark Sapir (Vanderbilt)

Title: On planar maps of non-positive curvature

Abstract: This is a joint work with A. Olshanskii. We prove that if a (4,4)-map M does not contain regular (d x d)-squares, then the area of M does not exceed Cdn where C is a constant, and n is the perimeter of M. Similar properties are proved for (6,3) and (3,6)-maps. Thus if a van Kampen diagram over a small cancelation presentation does not contain large regular subdiagrams, then the area of the diagram is small.

Wednesday, November 2, 2016

Stephen G. Simpson (Vanderbilt)

Title: Well partial orderings, with applications to algebra

Abstract: A partial ordering consists of a set P and a binary relation < on P which is transitive (x < y < z implies x < z) and irreflexive (x is never < x). Within P, a descending chain is a sequence a > b > c > ..., and an antichain is a set of elements a, b, c, ... which are pairwise incomparable (neither a < b nor b < a). A well partial ordering is a partial ordering which has no infinite descending chain and no infinite antichain. To each well partial ordering P one can associate an ordinal number o(P). For example, the natural numbers N with their usual ordering form a well ordering of order type omega and hence a well partial ordering with o(N) = omega. The class of well partial orderings is closed under finite sums, finite products, and certain other finitary operations. As noted in a 1972 paper by J. B. Kruskal, well partial ordering theory is a "frequently discovered concept" with many applications, especially in abstract algebra (G. Higman, I. Kaplansky, ...). As a simple example, Dickson's Lemma says that for each positive integer k the finite product N^k is a well partial ordering, and this is the key to a proof of the Hilbert Basis Theorem: for any field K and positive integer k, the polynomial ring K[x_1,...,x_k] has no infinite ascending chain of ideals. The ordinal number involved here is omega^omega. There are also generalizations involving larger ordinal numbers such as omega^{omega^omega}. There is a subclass of the well partial orderings, the better partial orderings, which has stronger closure properties. For example, if P is a better partial ordering, then the downwardly closed subsets of P form a better partial ordering under the subset relation. This fact from better partial ordering theory can be used to prove that for any field K, the group ring K[S] of the infinite symmetric group S (the direct limit of the finite symmetric groups S_n as n goes to infinity) has no infinite ascending chain of two-sided ideals and no infinite antichain of two-sided ideals. There seems to be an open question as to how far this theorem can be generalized from S to other locally finite groups. R. Laver has used better partial ordering theory to prove that the countable linear orderings form a well partial ordering (or rather, a well quasi-ordering) under the embeddability relation. N. Robertson and P. Seymour have proved a difficult theorem: the finite graphs form a well quasi-ordering under the minor embeddability relation. I. Kriz has proved that the Friedman trees are well quasi-ordered under the gap embeddability relation.

Wednesday, November 9, 2016

Ben Hayes (Vanderbilt)

Title: Metric approximations of wreath products

Abstract: I will discuss joint work with Andrew Sale. In it, we investigate metric approximations of wreath products. A mertic approximation of a group is a family of asymptotic homomorphisms into a class of groups so that the image of any nonidentity element is bounded away from zero. Metric approximations have received much recent interest and are related to several interesting conjectures, including Kaplansky's direct finiteness, Gottschalk's surjenctivity conjecture and the Connes embedding problem. Our results say the following: suppose that H is a sofic group. Then G wreath H is sofic (resp. linear sofic, resp. hyperlinear) if G is sofic (resp. linear sofic, resp. hyperlinear). No knowledge of sofic, linear sofic, or hyperlinear groups will be assumed.

Wednesday, November 16, 2016

Vito Zenobi (Universit of Montpellier 2)

Title: The tangent groupoid and secondary invariants in K-theory.

Abstract: I will explain how to define secondary invariants that detect exotic structures on smooth manifolds or metrics with positive scalar curvatures on Spin Riemannian manifolds. These invariants are elements in the K-theory of the tangent groupoid C*-algebra, introduced by Alain Connes to give a more conceptual viewpoint on index theory. These constructions easily generalize to more involved geometrical situations (such as foliations), well encoded by Lie groupoids.

Wednesday, November 30, 2016

Jing Tao (University of Oklahoma)

Title: Stable commutator lengths in right-angled Artin groups

Abstract: The commutator length of an element g in the commutator subgroup [G,G] of a group G is the smallest k such that g is the product of k commutators. When G is the fundamental group of a topological space, then the commutator length of g is the smallest genus of a surface bounding a homologically trivial loop that represents g. Commutator lengths are notoriously difficult to compute in practice. Therefore, one can ask for asymptotics. This leads to the notion of stable commutator length(scl) which is the speed of growth of the commutator length of powers of g. It is known that for n > 2, SL(n,Z) is uniformly perfect; that is, every element is a product of a bounded number of commutators, and hence scl is 0 on all elements. In contrast, most elements in SL(2,Z) have positive scl. This is related to the fact that SL(2,Z) acts naturally on a tree (its Bass-Serre tree) and hence has lots of nontrivial quasimorphisms. In this talk, I will discuss a result on the stable commutator lengths in right-angled Artin groups. This is a broad family of groups that includes free and free abelian groups. These groups are appealing to work with because of their geometry; in particular, each right-angled Artin group admits a natural action on a CAT(0) cube complex. Our main result is an explicit uniform lower bound for scl of any nontrivial element in any right-angled Artin group. This work is joint with Talia Fernos and Max Forester.

Wednesday, December 7, 2016

Gili Golan (Vanderbilt)

Title: Invariable generation of Thompson groups

Abstract: A subset S of a group G invariably generates G if for every choice of g(s) ∈ G,s ∈ S the set {sg(s):s ∈ S} is a generating set of G. We say that a group G is invariably generated if such S exists, or equivalently if S=G invariably generates G. In this talk, we study invariable generation of Thompson groups. We show that Thompson group F is invariable generated by a finite set, whereas Thompson groups T and V are not invariable generated. This is joint work with Tsachik Gelander and Kate Juschenko.

Wednesday, December 14, 2016

Andrew Putman (University of Notre Dame)

Title: The high-dimensional cohomology of the moduli space of curves with level structures

Abstract: We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the moduli space of curves is at least g-2. Well known conjectures of Looijenga would imply that this is sharp. This is joint work with Neil Fullarton.