Group Actions Seminar
(aka G mod Gamma seminar)

Schedule of Fall, 2008

• Sep 23 (Joint with Geometry/Topology Seminar): Hee Oh, Brown university
  
  Title: Counting Apollonian circles of bounded curvature
  
  Abstract: Apollonian circle packings are arrangements of tangent circles that arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. In a recent joint work with Alex Kontorovich, we obtain the asymptotic number of circles of curvature at most T, as T tends to infinity, in any gievn Apollonian packing. We also obtain the optimal upper bounds for the number of primes/twin primes in any integral Apollonian packing. In my talk, I will try to explain these results as well as to explain how the equidistribution of long horospheres in infinite volume hyperbolic 3-manifold are related.
  
  
• Oct 14: Dominique Hulin, Orsay University, France
  
  Iteration of quadrilateral foldings
  
  Abstract: We start with a quadrilateral q in the Euclidean plane. Replacing successively each of the four vertices by its symmetric with respect to the opposite diagonal, we obtain a new quadrilateral f(q), where f is our so-called ``cyclic folding transformation''. In a joint work with Y. Benoist, we study the dynamics of this transformation. This cyclic folding transformation was first studied by G. Darboux in 1879, who related it to translations on elliptic curves.
  
  
• Oct 21: Ludovic Marquis, Orsay University, France
  
  Properly convex projective surface of finite volume
  
  Abstract:
  
  
• Oct 28: Tsachik Gelander, Hebrew university, Israel
  
  On deformations of free groups in compact Lie groups.
  
  Abstract:I will discuss some properties of the varieties of deformations of free groups in compact Lie groups. In particular a proof of a conjecture of Margulis and Soifer about the density of nonvirtually free points, and a proof of a conjecture of Goldman on the ergodicity of the action of Aut(F_n) on such variety when n>2. I will also try to locate these result in a more general setup involving: finite groups, topological groups and actions of Aut(F_n), which recently re attract the interest of many people, and describe some of the main problems, and some recent developments.
  
  
• Nov 6 (Note the change of date): Fanny Kassel, Orsay University, France
  
  Clifford-Klein forms of semisimple rank-one group manifolds
  
  Abstract: Let G be a connected semisimple rank-one Lie group or a Zariski-connected semisimple rank-one algebraic group over a local field. I will describe all torsion-free discrete subgroups Gamma of G\times G acting properly discontinuously on G by left and right multiplication. I will also give a criterion on Gamma for the quotient to be compact, and discuss the deformation of such compact quotients.
  
  
• Nov 18 (Joint with Geometry/Topology Seminar): Alan Reid, University of Texas at Austin/IAS
  
  LERF and the Lubotkzy-Sarnak Conjecture
  
  Abstract: The Lubotzky-Sarnak Conjecture asserts that the fundamental group of a finite volume hyperbolic manifold does not have Property $\tau$. Put in a geometric context, this conjecture predicts a tower of finite sheeted covers for which the Cheeger constant goes to zero. This Conjecture has attracted a lot of attention recently because of its connections to the topology of finite sheeted covers of closed hyperbolic 3-manifolds. This talk will discuss these connections, together with recent work that connects these circle of ideas with the group theoretic property LERF (a far reaching generalization of residually finite).
  
  
• Nov 20 (Note the special date!): Jean Francois Quint, University of Paris 13, France
  
  Exponential drift and stationary measures on tori.
  
  Abstract:in a recent joint work with Yves Benoist, we developed a new method for the analysis of orbit structure for action of non-connected subgroups of Lie groups on homogeneous spaces. This method allowed us to prove that, if $\Gamma$ is a Zariski dense subgroup of $SL_d(\mathbb R)$, every $\Gamma$-orbit in $SL_d(\mathbb R)/SL_d(\mathbb Z)$, is either finite or dense. In this talk, I will show how this method can be used to recover the following result of Bourgain, Furman, Lindenstrauss and Mozes : if $\mu$ is a finitely supported probability measure on $SL_2(\mathbb Z)$ which support spans a Zariski dense subgroup of $SL_2(\mathbb R)$, then every extremal $\mu$-stationary Borel probability measure on the torus $T^2$ either is finitely supported or equals Lebesgue measure.
  
  
• Nov 25: Alex Gamburd, UCSC and Northwestern university
  (Alex will also be giving an Algebra seminar "Expanders: from arithmetic to combinatorics and back" on Nov 24)
  
  Title: Uniform spectral gap bounds
  
  Abstract:
  
  
  Past seminars
  
  Spring 08 Schedule
  Fall 07 Schedule
  
  
  
  
  
  
  
  
  
  Background poster is kindly provided by Rich Schwartz