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 TeichmÃ¼ller 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 is aspherical: roughly, they proved is a fiber bundle with locally homogeneous fibers. However, if 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. |
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 –manifold; the most interesting open case of this is for a hyperbolic –manifold. It’s also been asked — and answered — whether it holds if the target is or . I’ll say something about the wide range of proof strategies out there, and give a totally elementary construction to answer the 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 with Euclidean distance and Lebesgue measure . Let and . We say that the sequence of sees the space with speed if the set of points in infinitely many has full measure. In notation that is
Under the assumption that the Lebesgue Density Theorem implies this is equivalent to The Borel–Cantelli Theorem implies that fixing if any sequence sees the space with speed then . This talk classifies the sequences that see with speed for all nonincreasing sequences with divergent sum. The results extends to Ahlfors regular spaces. To motivate this, it is straightforward that for any fixed nonincreasing sequence with divergent sum we have that if are given by independent Lebesgue distributed random variables then almost surely sees with speed . In fact more is true. almost every sees the space with speed for all nonincreasing sequences with divergent sum. This is joint work with M. Boshernitzan. |
Preprint: Borel–Cantelli sequences Notes in pdf: Abstract Diophantine Equations |
November 22, 2012 is Thanksgiving. Happy holidays! |
December 6, 2012 |
Mostow Rigidity and Bounded Cohomology, Part 1 — Bena Tshishiku |
Mostow rigidity says that you can’t decorate a closed –manifold with more than one hyperbolic metric, unless . 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 |