Topology & Geometric Group Theory Seminar

Spring 2010

1:30 – 2:30, Malott 203

Tuesday, February 2

We study a new asymptotic invariant of a pair (A,G) where A is a subgroup of a group G, which we call commensurizer growth.

Roughly speaking, it counts the number of elements in the commensurizer of A in G with "index at most n", up to the action of the normalizer (denote these numbers by c_n(A,G)) and studies the asymptotic behavior of this sequence.

We compute commensurizer growth for several examples and find the following:

1. For the 3-dimensional Hiesenberg group H, c_n(H(Z),H(R))/n³ tends to 1/3ζ(3) as n tends to infinity, where ζ(x) is the Riemann zeta function.
2. We find a family of pairs (A,G) and (A,H) where while A is a uniform lattice with a dense commensurizer both in G and H, the commensurizer growths are very different: c_n(A,H) grows quadratically while c_n(A,G) grows exponentially.
3. Problem: is there a function f(n) such that for every lattice A in a finitely generated group G, c_n(A,G) < f(n) for all n?

This is joint work with Nir Avni and Seonhee Lim.