Group Actions Seminar

Brown Math Dept. Kassar 105
Roughly every other Thursday at 4-5pm

Contact: Hee Oh

Schedule (Fall, 2007)

• 9/27
  Alex Kontorovich, Brown university
  The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves
  
  abstract: We develop novel techniques using only abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through ``congruence'' subgroups. These methods have their origins in the work of Selberg, Lax-Phillips and Duke-Rudnick-Sarnak, and the uniformity relies on the spectral gap established in Bourgain-Gamburd and Bourgain-Gamburd-Sarnak. We give an application to the theory of affine linear sieves.
  
  
• 10/11
  Amir Mohammadi, Yale university
  Discrete Transitive Actions on Bruhat-Tits Building
  
  abstract: There are many discrete transitive actions on an $r$-regular tree, i.e. rank one Bruhat-Tits tree. However it is rare to have such action on a higher rank Bruhat-Tits Building. The objective of this joint on going work with "A. Salehi Golsefidy" is to classify all discrete transitive actions on Bruhat-Tits building of higher rank algebraic groups over characteristic zero local fields. In this talk we will see how arithmetic structure of such lattices and volume formula of Prasad for arithmetic lattices will give a non-existence result for such a lattice in $\rm {PGL_n} (F)$ for $n>8$ and give a list of 17 potential such examples for $4
  
  
• 10/25
  Alex Furman, UIC (also see his G-T talk on 10/24)
  Invariant and stationary measures for groups of toral automorphisms
  
  abstract: Joint work with J. Bourgain, E. Lindenstrauss, S. Mozes. Given a Zariski dense group G of toral automorphisms we prove that the only invariant or, more generally, stationary measures on the torus are combinations of Lebesgue and atomic measures.
  
  
• 11/15
  Emmanuel Breuillard, IAS and Polytechnique
  On the growth of Linear groups and Arithmetic heights
  
  abstract: We introduce a notion of minimal height for a finite subset of matrices with coefficients in an algebraic closure of Q and show an analog of the "Margulis Lemma" in this situation, which asserts that sets of small height must generate virtually solvable subgroups. This result allows to prove a uniform lower bound for the growth of finitely generated subgroups of GL_d(C), thus improving earlier results of Eskin-Mozes-Oh. We also make a connection with the Lehmer conjecture.
  
  
• 11/29
  Alireza Salehi Golsefidy, Princeton University
  Translates of horospherical measures and rational points on G/U
  
  abstract: (Joint with A. Mohammadi) There are three, more or less related, methods to understand distribution of integral or rational points on a homogeneous variety: Automorphic forms, mixing, and ergodic theory. In this talk, I will explain the general frame work of the second and the third methods. Then for the purpose of studying the distribution of rational points on a toric bundle on the flag variety, i.e. SL (n,R)/U where U is the set of real matrices with diagonal entries equal to 1, I will consider translates of the probability measure on SL(n,R)/SL(n,Z) which is induced by the Haar measure on U and supported on U SL(n,Z)/SL(n,Z). I will explain the possible scenarios for the limit of the translates, and show how it can help to study the distribution of the rational points.