Topology & Geometric Group Theory Seminar
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 cn(A,G)) and
studies the asymptotic behavior of this sequence.
We compute commensurizer growth for several examples and find the
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.
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.
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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