Fall 2019

Abstract: The Dehn function of a finitely presented group with finite generating set is a quantitative measure for the difficulty of detecting whether a word in represents the trivial element in . Dison raised the question if residually free groups admit a uniform polynomial upper bound on their Dehn functions. It is motivated by the existence of uniform polynomial upper bounds on interesting families of residually free groups, such as the Stallings–Bieri groups. In this talk we will show that the answer to Dison’s question is negative, by proving that for every there is a subgroup of a direct product of free groups with Dehn function bounded below by . This is joint work with Romain Tessera.

Abstract: Despite the simple looking presentation much is unknown about Artin groups. However, some questions can be answered in suitable classes of Artin groups. I will discuss the residual finiteness of Artin groups of large type on three generators. This relies on splittings of such Artin groups as amalgamated products of finitely generated free groups.

Abstract: The “virtual conjectures” in low-dimensional topology, stated by Thurston in 1982, postulated that every hyperbolic 3-manifold has finite covers that are Haken and fibered, with large Betti numbers. These conjectures were resolved in 2012 by Agol and Wise, using the machine of special cube complexes. Since that time, many mathematicians have asked how big a cover one needs to take to ensure one of these desired properties.

We begin to give a quantitative answer to this question, in the setting of alternating links in . If a prime alternating link has a diagram with crossings, we prove that the complement of has a special cover of degree less than . As a corollary, we bound the degree of the cover required to get Betti number at least . We also quantify residual finiteness, bounding the degree of a cover where a closed curve of length fails to lift. This is joint work with David Futer.

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.

Abstract: A (proper) cubulation of a group is a CAT(0) cube complex
together with a proper action of on . Partial cubulation drops
the properness assumption. Sometimes a partial cubulation can be
promoted to a proper cubulation. Sometimes a proper cubulation can be
improved by passing through a partial cubulation. I’ll give examples
of theorems of both types, focusing on hyperbolic and relatively
hyperbolic groups. This is joint work with Daniel Groves.

Abstract: Associated to any simplicial graph is the right-angled Coxeter group (RACG) whose presentation consists of an order 2 generator for each vertex of and relations stating that two generators commute if there is an edge between the corresponding vertices of . RACGs contain a rich class of subgroups including, up to commensurability, hyperbolic 3-manifold groups, surface groups, free groups, Coxeter groups and right-angled Artin groups to name a few. I will describe a procedure which associates a cube complex to a given subgroup of RACG. I will then present some results regarding structural and algorithmic properties of subgroups of RACGs whose proofs follow from this viewpoint. This is joint work with Pallavi Dani.

Abstract: For a given Riemann surface, there is a `uniformizing map’ from its universal cover to one of 3 simply connected complex manifolds, and each has a metric of constant curvature compatible with its complex structure.  Surfaces of negative Euler characteristic support hyperbolic metrics; we study the Teichmüller space of a surface by studying its hyperbolic metrics or its complex structures.  Analytically defined objects such as holomorphic differentials provide us with a rich source of (singular flat) geometry on our surface.  We present an approach for transporting data between the worlds of hyperbolic geometry and the singular flat geometry defined by quadratic differentials, extending a result of Mirzakhani from a measurably generic setting to one of full generality.  This is joint work with Aaron Calderon.

Abstract: TBA