Spring 2020

Abstract: (Joint with Matt Haulmark)
A JSJ decomposition is a graph of groups decomposition that allows one to classify all splittings of a group over certain subgroups. I will discuss a JSJ decomposition for relatively hyperbolic groups splitting over elementary subgroups that depends only on the topology of its boundary. This decomposition could potentially be of use for understanding groups that have homeomorphic boundaries, but are not necessarily quasi-isometric.

Abstract: The Dehn function is a measure of complexity of the word problem for a finitely presented group. In the class of almost finitely presented groups there is another well-defined function called the homological Dehn function. I will introduce the class of almost finitely presented groups and Dehn functions. I will finish by discussing recent joint work with Noel Brady and Ignat Soroko showing that the class of almost finitely presented groups acts very differently from the class of finitely presented groups.

Abstract: Many quasi-isometry invariants of groups and spaces are the result of translating classical topological invariants into the large-scale world. A seminal example of this is asymptotic dimension, which is a coarse analog of Lebesgue covering dimension. In this talk I will look at a recently introduced notion of coarse homotopy and coarse homotopy groups. I will prove a Coarse Lifting Lemma, which allows us to compute fundamental groups of spaces with interesting large scale structure, such as warped cones.

Abstract: Given a geodesically complete CAT(-1) space, we can associate to it a boundary at infinity which can be equipped with a cross ratio. It is well known that the boundary, together with the cross ratio contains a lot of information about the geometry of the interior space. One way to make this relationship explicit uses a construction called the circumcenter extension. It turns out that this construction can be generalized to a large class of CAT(0) spaces and still provide interesting results. In this talk, we will survey known results about the circumcenter extension, show how to construct the circumcenter extension and outline how to obtain results from it.

Abstract: We will define the rank one symmetric spaces using their Lie group structure, then examine a class of Lie groups which serve as a generalization of these well known metric spaces. By examining the relationship between the metric and Lie structures on hyperbolic, complex hyperbolic, and quaternionic hyperbolic spaces, we can get an idea of natural conditions for metrics on the general class of Heintze spaces of Carnot type. We prove the existence of metrics that satisfy a pinched curvature condition, and show that in a special case this pinching is optimal.

Abstract: Let Phi be a finite root system. A Phi-grading of a group G is a family of subgroups of G satisfying certain axioms that are inspired by the family of root subgroups in a Chevalley group.

In the 1990s Shi showed that the root subgroups of Phi-grading are coordinatized by an associative ring if Phi is simply laced, irreducible and of rank at least three. For the remaining types there are only partial results available.

In my talk I give an introduction to Phi-gradings of groups and present some recent results about them. To a large part they are motivated by the theory of buildings and applications to rank one groups of exceptional type.

This is joint work with Richard Weiss.

Abstract: Square-tiled surfaces (STSs) are branched covers of the standard square torus with branching over exactly one point. In this talk we consider a randomizing model for STSs and generalizations to random branched covers of other simple translation surfaces. We will then see how to use the model to obtain some topological and geometric statistics of these surfaces.

Abstract: In this talk, we will discuss a class of sub-Finsler metrics on the real Heisenberg group H(R), arising as asymptotic cones of word metrics on the integer Heisenberg group H(Z) for various generating sets. We will describe new results on boundaries for these Carnot-Carthéodory metrics, with applications to the study of random walks on H(Z). This is joint work with Sebastiano Nicolussi Golo.

Abstract: It is a classical fact that every finitely generated group embeds as a subgroup of a finitely generated simple group. In the 90’s Bridson proved that if one relaxes “simple” to “no proper finite index subgroups” then such an embedding can be done in a quasi-isometric way. In joint work with Jim Belk, we prove that this is true even keeping the word “simple”: every finitely generated group quasi-isometrically embeds as a subgroup of a finitely generated simple group. The simple groups we construct are “twisted” variants of Brin-Thompson groups. Certain of these twisted Brin-Thompson groups also provide examples of groups with interesting finiteness properties, and using them we can produce the second known family of simple groups with arbitrary finiteness properties (the first being due to Skipper-Witzel-Z).

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