Topology & Geometric Group Theory Seminar

Spring 2010

1:30 – 2:30, Malott 203

Tuesday, February 2

Eran Nevo, Cornell University

Commensurizer growth

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 cn(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, cn(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: cn(A,H) grows quadratically while cn(A,G) grows exponentially.
  3. Problem: is there a function f(n) such that for every lattice A in a finitely generated group G, cn(A,G) < f(n) for all n?

This is joint work with Nir Avni and Seonhee Lim.

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