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.

Oct 30 Speaker
Institution
  TBA

  Abstract:

TBA

Nov 6

2:00 pm

Julia Plavnik
Texas A&M
  TBA

  Abstract:

TBA

Notice that we will be meeting in a different room and at a different time: APM 7218

Nov 14, Tue

10:00 am

Wenyu Pan
Yale University
  TBA

  Abstract:

TBA

Notice that we will be meeting in a different room and at a different time: APM 6402

Nov 20 Speaker
Institution
  TBA

  Abstract:

TBA

Notice that we will be meeting in a different room: APM ??

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.

Dec 4 Speaker
Institution
  TBA

  Abstract:

TBA