Thursday, September 28, 2017 at 3pm in University Hall (UH) 28
Mike Davis, Ohio State University
Action dimensions of simple complexes of groups
Abstract
The geometric dimension of a torsion-free group G is the minimum dimension of a model for BG by a CW complex. Its action dimension is the minimum dimension of a model for BG by a manifold. I will discuss recent work with Kevin Schreve and Giang Le in which we compute the action dimension of Artin groups, graph products and other examples of groups which are given as simple complexes of groups.
Tuesday, October 3, 2017 at 3pm in Cockins Hall (CH) 240 (special day and location)
Thomas Koberda, University of Virginia
Free products and diffeomorphisms of compact manifolds
Abstract
It is a well-known fact that if G and H are groups of homeomorphisms of the interval or of the circle, then the free product G*H is also a group of homeomorphisms of the interval or of the circle, respectively. I will discuss higher regularity of group actions, showing that if G and H are groups of C∞ diffeomorphisms of the interval or of the circle, then G*H may fail to act by even C2 diffeomorphisms on any compact one-manifold. As a corollary, we can classify the right-angled Artin groups which admit faithful C2 actions on the circle, and recover a joint result with H. Baik and S. Kim. This is joint work with S. Kim.
Thursday, October 19, 2017 at 3pm in University Hall (UH) 28
Mark Pengitore, Purdue University
Translation-like actions of nilpotent groups
Abstract
Whyte introduced translation-like actions of groups which serve as a geometric generalization of subgroup containment. He then proved a geometric reformulation of the von Neumann conjecture by demonstrating a finitely generated group is nonamenable if and only if it admits a translation-like action by a non-abelian free group. This provides motivation for the study of what groups can act translation-like on other groups. As a consequence of Gromov's polynomial growth theorem, only nilpotent groups can act translation-like on other nilpotent groups. In joint work with David Cohen, we demonstrate that if two nilpotent groups have the same growth, but non-isomorphic Carnot completions, then they can't act translation-like on each other.
Thursday, November 2, 2017 at 3pm in University Hall (UH) 28
Sahana Balasubramanya, Vanderbilt University
Acylindrical group actions on quasi-trees
Abstract
A group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group G has a generating set X such that the corresponding Cayley graph is a (non-elementary) quasi-tree and the action of G on the Cayley graph is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As a by-product, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups.
Thursday, November 16, 2017 at 3pm in University Hall (UH) 28
Kevin Schreve, University of Michigan
Action dimension and L2-cohomology
Abstract
The action dimension of a group G is the minimal dimension of contractible manifold that G acts on properly discontinuously. Conjecturally, if a group has nontrivial L2 cohomology in dimension n, the action dimension of G is bounded below by 2n. I will discuss this conjecture for graph products of fundamental groups of aspherical manifolds and fundamental groups of complex hyperplane complements. This is joint work with Mike Davis and Giang Le.
Thursday, November 30, 2017 at 3pm
Matthew Durham, Yale University
Geometrical finiteness and Veech subgroups of mapping class groups
Abstract
I will discuss work in progress with Dowdall, Leininger, and Sisto, in which we aim to develop a notion of geometrical finiteness for subgroups of mapping class groups. Motivated by the theory of convex cocompact subgroups, which are precisely those which determine hyperbolic surface group extensions, I will describe some hyperbolic properties of the surface group extensions coming from lattice Veech subgroups.
Tuesday, February 6, 2018 at 1:50pm in Denney Hall (DE) 265
Christoforos Neofytidis, University of Geneva
Aspherical circle bundles and a problem of Hopf
Abstract
A long standing question of Hopf asks whether every self-map of absolute degree one of a closed oriented manifold is a homotopy equivalence. This question gave rise to several other problems, most notably whether the fundamental groups of aspherical manifolds are Hopfian, i.e. any surjective endomorphism is an isomorphism. Recall that the Borel conjecture states that any homotopy equivalence between two closed aspherical manifolds is homotopic to a homeomorphism. In this talk, we verify a strong version of Hopf's problem for certain aspherical manifolds. Namely, we show that a self-map of a circle bundle over a closed oriented negatively curved manifold is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is trivial. Our main result is that a non-trivial circle bundle over a closed oriented negatively curved manifold does not admit self-maps of absolute degree greater than one. This extends in all dimensions the case of circle bundles over closed hyperbolic surfaces (which was shown by Brooks and Goldman in their study of the Seifert volume) and provides the first examples of non-vanishing semi-norms on the fundamental classes of circle bundles over negatively curved manifolds in all dimensions.
Thursday, February 22, 2018 at 1:50pm in Denney Hall (DE) 265
Lorenzo Ruffoni, University of Bologna/Yale University
Geometric structures on surfaces with a maximal number of symmetries
Abstract
One way in which a geometric structure on a manifold is more rigid than just a topological structure is exemplified by the fact that the group of geometric automorphisms is in general smaller than the group of self-homeomorphisms. As shown by Hurwitz in 1892, complex structures on surfaces of genus at least 2 are rigid enough to guarantee that the group of holomorphic symmetries is finite, with a cardinality bounded just in terms of the genus. Structures which achieve this bound, and their groups of symmetries, are known to enjoy special geometric and algebraic properties. Motivated by this classical theory, and by recent work on translation surfaces by Schlage-Puchta and Weitze-Schmithüsen, we consider Hurwitz-like problems for complex projective structures on surfaces. In a sense that will be made precise, the most symmetric structures turn out to be the uniformizations of Galois Belyi curves by Fuchsian triangle groups. This is a joint work with G. Faraco.
Thursday, April 5, 2018
Rita Gitik, University of Michigan
A new algorithm in group theory
Abstract
We describe a new algorithm which determines if the intersection of a quasiconvex subgroup of a negatively curved group with any of its conjugates is infinite. The algorithm is based on the concepts of a coset graph and a weakly Nielsen generating set of a subgroup. We also give a new proof of decidability of a membership problem for quasiconvex subgroups of negatively curved groups.