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 c_{n}(A,G)) and
studies the asymptotic behavior of this sequence.
We compute commensurizer growth for several examples and find the
following:

For the 3dimensional Hiesenberg group H,
c_{n}(H(Z),H(R))/n^{3} tends to 1/3ζ(3) as n tends to
infinity, where ζ(x) is the Riemann zeta function.

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.

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.
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