Title and Abstracts: WTC 2010

Aaron Abrams, Emory

Title: Quasi-isometric rigidity of graph braid groups

Abstract: We show that the size of a complete graph is determined by the quasi-isometry type of its 2-string braid group. This is joint work with Praphat Fernandes.

Khalid Bou-Rabee, University of Chicago

Title: Quantifying Residual Finiteness of Arithmetic Groups

Abstract: We begin with an introduction to the quantification of residual finiteness. This program will lead us to define the "normal Farb growth" of a group, which quantifies how well-approximated the group is by its finite quotients. Our main goal will be to present a result of ours that any $S$-arithmetic subgroup of a higher rank Chevalley group $G$ has normal Farb growth $n^{\dim(G)}$. We will focus on the examples $G = SL(n,C)$, where $n > 2$. We will also work out a couple of examples to indicate why our result fails for non-linear and non-arithmetic examples. This talk covers joint work with Tasho Kaletha.

Spencer Dowdall, University of Chicago

Title: Dilatation vs self-intersection number for point-pushing pseudo-Anosovs

Abstract: This talk is about the dilatations of pseudo-Anosov mapping classes obtained by pushing a marked point around a filling curve. After reviewing this "point-pushing" construction, I will give both upper and lower bounds on the dilatation in terms of the self-intersection number of the filling curve. The upper bounds involve analyzing explicit examples using train tracks, and the lower bound is obtained by lifting to the universal cover and studying the images of simple closed curves. As a corollary, we also bound the least dilatation of any pseudo-Anosov in the point-pushing subgroup.

Søren Galatius, Stanford University

Title: Madsen-Weiss for geometrically minded topologists.

Abstract: This will be a series of three lectures. My talks will be based on my joint paper with Eliashberg and Mishachev (arxiv), where we give a proof of Madsen-Weiss' generalized Mumford conjecture. Compared with the original proof, our proof is intended to be more geometric and less homotopy theoretic.

Lars Louder, University of Michigan

Title: Nielsen equivalence of generating sets for surface groups.

Abstract: We will show that generating sets for surface groups, except for the connected sum of three projective planes, are either reducible or Nielsen equivalent to standard generating sets, improving upon a theorem of Zieschang. This is equivalent to the statement that Aut(F_n) acts transitively on Epi(F_n,S) when S is a surface group.

Mark Meilstrup, Brigham Young University

Title: Some Results in Wild Low-Dimensional Topology and Dynamics

Abstract: This talk discusses spaces that are wild, i.e. not locally simply connected. We first discuss periodic properties of maps from a given space to itself, similar to Sharkovsky's Theorem for interval maps. We study many non-locally connected spaces and show that some have periodic structure either identical or related to Sharkovsky's result, while others have essentially no restrictions on the periodic structure. We next consider embeddings of solenoids together with their complements in three space. We differentiate solenoid complements via both algebraic and geometric means, and show that every solenoid has an unknotted embedding with Abelian fundamental group, as well as infinitely many inequivalent knotted embeddings with non-Abelian fundamental group.

Jason Manning, University at Buffalo, SUNY

Title: Nonpositively curved realizations of group-theoretic Dehn fillings

Abstract: Group-theoretic "Dehn filling" gives a method of obtaining new (relatively) hyperbolic groups as "children" of old ones. In case the parent has some stronger geometric properties, we'd like to know if the children share these properties. We give some specific examples where the quotient groups "inherit" some CAT(0)/CAT(-1) geometry from their parents. We use this geometry to get extra information about the children not apparent from coarse versions of the Dehn filling theorem. Some of the work described is joint with Koji Fujiwara and some is joint with Daniel Groves.

Title: Quasi-isometric rigidity of graph braid groups

Abstract: We show that the size of a complete graph is determined by the quasi-isometry type of its 2-string braid group. This is joint work with Praphat Fernandes.

Khalid Bou-Rabee, University of Chicago

Title: Quantifying Residual Finiteness of Arithmetic Groups

Abstract: We begin with an introduction to the quantification of residual finiteness. This program will lead us to define the "normal Farb growth" of a group, which quantifies how well-approximated the group is by its finite quotients. Our main goal will be to present a result of ours that any $S$-arithmetic subgroup of a higher rank Chevalley group $G$ has normal Farb growth $n^{\dim(G)}$. We will focus on the examples $G = SL(n,C)$, where $n > 2$. We will also work out a couple of examples to indicate why our result fails for non-linear and non-arithmetic examples. This talk covers joint work with Tasho Kaletha.

Spencer Dowdall, University of Chicago

Title: Dilatation vs self-intersection number for point-pushing pseudo-Anosovs

Abstract: This talk is about the dilatations of pseudo-Anosov mapping classes obtained by pushing a marked point around a filling curve. After reviewing this "point-pushing" construction, I will give both upper and lower bounds on the dilatation in terms of the self-intersection number of the filling curve. The upper bounds involve analyzing explicit examples using train tracks, and the lower bound is obtained by lifting to the universal cover and studying the images of simple closed curves. As a corollary, we also bound the least dilatation of any pseudo-Anosov in the point-pushing subgroup.

Søren Galatius, Stanford University

Title: Madsen-Weiss for geometrically minded topologists.

Abstract: This will be a series of three lectures. My talks will be based on my joint paper with Eliashberg and Mishachev (arxiv), where we give a proof of Madsen-Weiss' generalized Mumford conjecture. Compared with the original proof, our proof is intended to be more geometric and less homotopy theoretic.

Lars Louder, University of Michigan

Title: Nielsen equivalence of generating sets for surface groups.

Abstract: We will show that generating sets for surface groups, except for the connected sum of three projective planes, are either reducible or Nielsen equivalent to standard generating sets, improving upon a theorem of Zieschang. This is equivalent to the statement that Aut(F_n) acts transitively on Epi(F_n,S) when S is a surface group.

Mark Meilstrup, Brigham Young University

Title: Some Results in Wild Low-Dimensional Topology and Dynamics

Abstract: This talk discusses spaces that are wild, i.e. not locally simply connected. We first discuss periodic properties of maps from a given space to itself, similar to Sharkovsky's Theorem for interval maps. We study many non-locally connected spaces and show that some have periodic structure either identical or related to Sharkovsky's result, while others have essentially no restrictions on the periodic structure. We next consider embeddings of solenoids together with their complements in three space. We differentiate solenoid complements via both algebraic and geometric means, and show that every solenoid has an unknotted embedding with Abelian fundamental group, as well as infinitely many inequivalent knotted embeddings with non-Abelian fundamental group.

Jason Manning, University at Buffalo, SUNY

Title: Nonpositively curved realizations of group-theoretic Dehn fillings

Abstract: Group-theoretic "Dehn filling" gives a method of obtaining new (relatively) hyperbolic groups as "children" of old ones. In case the parent has some stronger geometric properties, we'd like to know if the children share these properties. We give some specific examples where the quotient groups "inherit" some CAT(0)/CAT(-1) geometry from their parents. We use this geometry to get extra information about the children not apparent from coarse versions of the Dehn filling theorem. Some of the work described is joint with Koji Fujiwara and some is joint with Daniel Groves.