University of California, San Diego. Academic year: 2017-2018; Fall. Monday 3:00-4:00. APM 7421.
 Date Speaker Topic Oct 23 David El-Chai Ben-Ezra UCSD The Congruence Subgroup Problem for $${\rm Aut}(F_2)$$   Abstract: The classical congruence subgroup problem asks whether every finite quotient of $$G={\rm GL}_{n}\left(\mathbb{Z}\right)$$ comes from a finite quotient of $$\mathbb{Z}$$. I.e. whether every finite index subgroup of $$G$$ contains a principal congruence subgroup of the form $$G\left(m\right)=\ker\left(G\to {\rm GL}_{n}\left(\mathbb{Z}/m\mathbb{Z}\right)\right)$$ for some $$m\in\mathbb{N}$$? If the answer is affirmative we say that $$G$$ has the congruence subgroup property (CSP). It was already known in the $$19^{\underline{th}}$$ century that $${\rm GL}_{2}\left(\mathbb{Z}\right)$$ has many finite quotients which do not come from congruence considerations. Quite surprising, it was proved in the sixties that for $$n\geq3$$, $${\rm GL}_{n}\left(\mathbb{Z}\right)$$ does have the CSP. Observing that $${\rm GL}_{n}\left(\mathbb{Z}\right)\cong {\rm Aut}\left(\mathbb{Z}^{n}\right)$$, one can generalize the congruence subgroup problem as follows: Let $$\Gamma$$ be a group. Does every finite index subgroup of $$G={\rm Aut}\left(\Gamma\right)$$ contain a principal congruence subgroup of the form $$G(M)=\ker\left(G\to {\rm Aut}\left(\Gamma/M\right)\right)$$ for some finite index characteristic subgroup $$M\leq\Gamma$$? Very few results are known when $$\Gamma$$ is not abelian. For example, we do not know if $${\rm Aut}\left(F_{n}\right)$$ for $$n\geq3$$ has the CSP. But, in 2001 Asada proved, using tools from algebraic geometry, that $${\rm Aut}\left(F_{2}\right)$$ does have the CSP, and later, Bux-Ershov-Rapinchuk gave a group theoretic version of Asada's proof (2011). On the talk, we will give an elegant proof to the above theorem, using basic methods of profinite groups and free groups. Nov 3, Friday 1:00 pm Julia Plavnik Texas A&M Projectivity of tensor products for some Hopf algebras   Abstract: In this talk, we will pose some questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. To give some answers to these questions, we will construct some examples coming from smash copropucts of Sweedler Hopf algebras. One of the fundamental tools that we use to understand the modules of these Hopf algebras is the theory of support varieties. If time allows, we will mention the definition and some of the main properties of the support varieties for these examples. Notice that we will be meeting in a different room and at a different time: APM 7218 Nov 14, Tuesday 10:00 am Wenyu Pan Yale University Local mixing and abelian covers of finite volume hyperbolic manifolds   Abstract: Abelian covers of finite volume hyperbolic manifolds are ubiquitous. We will discuss ergodic properties of the geodesic flow/ frame flow on such spaces. In particular, we will discuss the local mixing property of the geodesic flow/ frame flow, which we introduce to substitute the well-known strong mixing property in infinite volume setting. We will also discuss applications to measure classification problems and to counting and equidistribution problems. Part of the talk is based on the joint work with Hee Oh. Notice that we will be meeting in a different room and at a different time: APM 6402 Nov 27 Francois Thilmany UCSD Lattices of minimal covolume in $${\rm SL}_n(\mathbb{R})$$   Abstract: A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $$\mathrm{SL}_2(\mathbb{R})$$. In general, given a semisimple Lie group $$G$$ over some local field $$F$$, one may ask which lattices in $$G$$ attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel's result, Lubotzky determined that a lattice of minimal covolume in $$\mathrm{SL}_2(F)$$ with $$F=\mathbb{F}_q(\!(t)\!)$$ is given by the so-called the characteristic $$p$$ modular group $$\mathrm{SL}_2(\mathbb{F}_q[1/t])$$. He noted that, in contrast with Siegel's lattice, the quotient by $$\mathrm{SL}_2(\mathbb{F}_q[1/t])$$ was not compact, and asked what the typical situation should be: for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice? In the talk, we will review some of the known results, and then discuss the case of $$\mathrm{SL}_n(\mathbb{R})$$ for $$n > 2$$. It turns out that, up to automorphism, the unique lattice of minimal covolume in $$\mathrm{SL}_n(\mathbb{R})$$ is $$\mathrm{SL}_n(\mathbb{Z})$$. In particular, it is not uniform, giving an answer to Lubotzky's question in this case.