# Fall 2012

 October 4, 2012 Action Dimension of Discrete Groups — Grigori Avramidi It is well known that the dimension of the smallest aspherical complex with a given fundamental group is the homological dimension of that group (except possibly for homological dimension $2$). But, what is the smallest dimension of an aspherical manifold (usually noncompact) with a given fundamental group? Bestvina and Feighn showed that a lattice in a symmetric space is not the fundamental group of an aspherical manifold of lower dimension. One also knows the answer for the mapping class group, Out$( F_n )$ and some right angled Artin groups. The goal of this talk is to explain where these types of results come from.
 October 11, 2012 Realization Problems in the Mapping Class Group — Nick Salter This is a practice talk for my topic exam. The mapping class group of a surface is defined as the group of isotopy classes of diffeomorphisms. In this talk, we will discuss when a group of mapping classes can be realized as diffeomorphisms; we will have occasion to draw on Teichmüller theory, characteristic classes of surface bundles, and flat circle bundles. In the affirmative direction, a theorem of Nielsen shows that finite cyclic groups are realizable; Kerckhoff extended this to all finite groups. In the opposite direction, Morita showed that there is no lift of the entire mapping class group. If time permits, we will also present a result of Bestvina–Church–Souto which proves the non-lifting of an infinite-index subgroup and gives another proof of Morita’s result. Since this is a topic practice talk, questions are strongly encouraged. Notes in pdf: Realization Problems in the Mapping Class Group
 October 18, 2012 Riemannian Manifolds with Nontrivial Local Symmetry — Wouter van Limbeek In this talk I will discuss the problem of classifying all closed Riemannian manifolds whose universal cover has nondiscrete isometry group. Farb and Weinberger solved this under the additional assumption that $M$ is aspherical: roughly, they proved $M$ is a fiber bundle with locally homogeneous fibers. However, if $M$ is not aspherical, many new examples and phenomena appear. I will exhibit some of these, and discuss progress towards a classification. One new ingredient is Frankel’s technique of averaging a harmonic map, which may be of independent interest. Notes in pdf: Riemannian Manifolds with Nontrivial Local Symmetry
 October 23, 2012 Note: FFSS is on Tuesday this week, beginning at 3pm. The room is as usual, E308. Variations on the Simple Loop Conjecture — Katie Mann The simple loop conjecture (now a theorem thanks to Gabai) says that any non-injective homomorphism from a closed surface group to another closed surface group has an element represented by a simple closed curve in the kernel. It’s been conjectured that the result still holds if the target is replaced by the fundamental group of an orientable $3$–manifold; the most interesting open case of this is for a hyperbolic $3$–manifold. It’s also been asked — and answered — whether it holds if the target is $\mathrm{PSL}(2,\mathbb{C})$ or $\mathrm{PSL}(2,\mathbb{R})$. I’ll say something about the wide range of proof strategies out there, and give a totally elementary construction to answer the $\mathrm{PSL}$ case. Preprint: A counterexample to the simple loop conjecture for PSL(2,R) Notes in pdf: Variations on the Simple Loop Conjecture
 October 30, 2012 Note: FFSS this week is on Tuesday, starting at 3pm, in the usual room E308. Abstract Diophantine Approximation — Jon Chaika Consider the unit interval $[0,1]$ with Euclidean distance $d$ and Lebesgue measure $\ensuremath{\lambda}$. Let $p_1,p_2,... \subset [0,1]$ and $a_1>a_2>...\subset \mathbb{R}^+$. We say that the sequence of $\{p_i\}$ sees the space with speed$\{a_i\}$ if the set of points in infinitely many $B(p_i,a_i)$ has full measure. In notation that is $\ensuremath{\lambda}(\ensuremath{\underset{n=1}{\overset{\infty}{\cap}} \, {\underset{i=n}{\overset{\infty}{\cup}}}\,} B(p_i,a_i))=1.$ Under the assumption that $\underset{i \to \infty}{\lim}a_i=0$ the Lebesgue Density Theorem implies this is equivalent to $\ensuremath{\lambda}(\{x:\underset{i \to \infty}{\lim}\mathrm{inf} \; \; a_i^{-1}d(p_i,x)=0\})=1.$ The Borel–Cantelli Theorem implies that fixing $\{a_i\}$ if any sequence sees the space with speed $\{a_i\}$ then $\sum_{i=1}^{\infty} a_i=\infty$. This talk classifies the sequences $\{p_i\}$ that see $[0,1]$ with speed $\{a_i\}$ for all nonincreasing sequences $\{a_i\}$ with divergent sum. The results extends to Ahlfors regular spaces. To motivate this, it is straightforward that for any fixed nonincreasing sequence $\{a_i\}$ with divergent sum we have that if $\{p_i\}$ are given by independent Lebesgue distributed random variables then $\ensuremath{\lambda}^{\mathbb{N}}$ almost surely $\{p_i\}$ sees $[0,1]$ with speed $\{a_i\}$. In fact more is true. $\ensuremath{\lambda}^{\mathbb{N}}$ almost every $\{p_i\}$ sees the space with speed $\{a_i\}$ for all nonincreasing sequences $\{a_i\}$ with divergent sum. This is joint work with M. Boshernitzan. Preprint: Borel–Cantelli sequences Notes in pdf: Abstract Diophantine Equations
 November 7, 2012 Note: FFSS this week is on Wednesday at 3pm, in E207. FI–Modules and Representations of the Classical Weyl Groups — Jenny Wilson Earlier this year, Church, Ellenberg, and Farb introduced FI–modules, a tool for studying sequences of representations of the symmetric groups. I will review the definition and theory of FI–modules, and describe how they generalize to sequences of representations of the classical Weyl groups in Type B/C and D. The theory of FI–modules has provided a wealth of new results by numerous authors in algebra, geometry, and topology. I will outline some of these results, including applications to configurations spaces and the pure string motion group. This is practice for a talk at the University of Wisconsin, Madison.
 November 15, 2012 SL(2,R) Orbit Closures of Translation Surfaces — Alex Wright Over the past three decades, it has been discovered that the $\mathrm{SL}(2,\mathbb{R})$ orbit closure of a translation surface governs, to a surprising extent, properties of the translation surface. This includes counting problems, the dynamics of straight line flow, and flat geometry. However, until now, the orbit closure was known only for relatively few translation surfaces. We will outline a proof that any translation surface with sufficiently transcendental period coordinates has orbit closure as large as possible. We will also discuss what is known and conjectured about the global structure of orbit closures.
 November 22, 2012 is Thanksgiving. Happy holidays!
 November 29, 2012 Commensurations of Arithmetic Groups — Daniel Studenmund Roughly speaking, an arithmetic group is the set of integer matrices inside an algebraic matrix group. A simple example is $\mathbb{Z}^n$ inside $\mathbb{R}^n$, where $\mathbb{R}^n$ is a unipotent subgroup of $\mathrm{GL}(n+1, \mathbb{R})$. The automorphism group of $\mathbb{R}^n$ is $\mathrm{GL}(n, \mathbb{R})$, while the abstract commensurator group of $\mathbb{Z}^n$ is $\mathrm{GL}(n, \mathbb{Q})$. In this talk I will state a generalization to describe abstract commensurators for arithmetic groups which satisfy strong rigidity, and state partial results for nonrigid groups. There will be definitions and examples.
 December 6, 2012 Mostow Rigidity and Bounded Cohomology, Part 1 — Bena Tshishiku Mostow rigidity says that you can’t decorate a closed $n$–manifold with more than one hyperbolic metric, unless $n=2$. In the 80’s Gromov gave a new proof of this theorem using simplicial volume and measure homology. Recently, Bucher–Burger–Iozzi showed that one may replace measure homology with bounded cohomology to give yet another proof. I will describe some of their methods by proving the following theorem of Gromov: The simplicial volume and the volume of a closed hyperbolic manifold are proportional, and the proportionality constant depends only on the dimension. Notes in pdf: Mostow Rigidity and Bounded Cohomology, Part 1