Fall 2020 The Zoom Semester

Abstract:  Minimal volume entropy is a Riemannian manifold invariant introduced by Gromov to study the asymptotic geometry of the universal cover, and is related to the growth of the fundamental group.  Here we study an extension of this invariant to a simplicial complex X equipped with a piecewise Riemannian metric.  When X is 2-dimensional and aspherical, we characterize when the minimal volume entropy vanishes in terms of an algebraic condition on the fundamental group.  We apply these results to the cases of free-by-cyclic groups and 2-dimensional right-angled Artin groups, obtaining a uniform lower bound in the nonvanishing case.  This is joint work with Matt Clay.

Abstract: A group is said to have property (T) if any action on a Hilbert space, has a fixed point. I will discuss what happens to this notion when one replaces Hilbert space with a more general Lp space, and discuss in particular the case of (relatively) hyperbolic groups.

Abstract: Studying the topology of a space “at infinity” offers a powerful perspective in geometric group theory. To capture the structure at infinity relative to a subspace, one can consider relative ends and associated deep homology groups. A goal is then to compute these homology groups and use homological arguments to study these pairs of spaces. In this talk, I will explain how both goals are possible in the setting of coarse embeddings into coarse PD(n) spaces. As applications, we use homological arguments to prove that certain groups cannot act properly on a given manifold. This is joint work with Chris Hruska and Hung Cong Tran.

Abstract: [Circle bundles over a surface have an “Euler number,” which is the
obstruction to finding a continuous section of the bundle.  For flat circle
bundles, the Euler number is bounded; this is known as the Milnor—Wood
inequality.  I will define flat circle bundles and the Euler number of a
bundle.  Using the Milnor—Wood inequality, we will give a short and clever
proof, due to Bestvina—Church—Souto, that the point pushing subgroup of the
mapping class group of a closed surface with one marked point cannot be
realized as a group of diffeomorphisms fixing the marked point.

Abstract: Hongbin Sun showed that a closed hyperbolic 3-manifold virtually contains any prescribed torsion subgroup as a direct factor in homology. In this talk we will discuss joint work with Daniel Groves generalizing Sun’s result to irreducible 3-manifolds which are not graph-manifolds.

Abstract: I will discuss some results and questions about the behavior, with respect to algebraic fibrations, for the classes of groups above, using as reference point the fundamental groups of Kodaira fibrations, that belong to both.

Abstract:

Abstract: We discuss arithmeticity of groups G(A) = ℤ^n ⋊ ℤ, where ℤ acts on ℤ^n by powers of an irreducible, hyperbolic matrix A ∈ GL(n,ℤ). The question of when G(A) is arithmetic was studied systematically by Grunewald-Platonov, but there are basic things that we still don’t know. For example, for what values of n is there an arithmetic example? We discuss some progress toward answering this question. 

The Dehn function of a finitely presented group is a classical invariant to study the complexity of the word problem. One can study the homological analogue of this function, namely, the homological filling function. This function is defined on the larger class of almost finitely presented groups. However, there has not been much study of the homological filling function of almost finitely presented groups. I will define all the relevant notions and discuss joint work with Noel Brady and Ignat Soroko studying these functions.

I will explain a complicated system of recursive polynomials which describe the hyperbolic structures on the complements of 2-bridge links.  The patterns in this system of polynomials can be used to efficiently compute these geometries and lend themselves to straightforward inductive proofs.

Abstract: TBA