On groups with locally compact asymptotic cones

We show how a recent result of Hrushovsky [6] implies that if an asymptotic cone of a finitely generated group is locally compact, then the group is virtually nilpotent. Let G be a group generated by a finite set X. Then G can be considered as a metric space where dist(g, h) is the length of a shortest word on X ∪X−1 representing g−1h. Let ω be a non-principal ultrafilter on N, i.e. a function from the set of subsets P (N) to {0, 1} with ω(A ∪B) = ω(A) + ω(B) if A ∩B = ∅, ω(N) = 1 and ω(A) = 0 for every finite set A. Note that if A ⊆ N, then ω(A) + ω(N \ A) = 1 because N is a disjoint union of A and N \A. For every A ⊆ B ⊆ N, we have ω(A) ≤ ω(B) because B is a disjoint union of B \A and A. For every two subsets A,B ⊆ N, ω(A ∪ B) ≤ ω(A) + ω(B) because A ∪ B is a disjoint union of A \B ⊆ A and B. Also if ω(A) = ω(B) = 1, then ω(A ∩ B) = 1 because A ∩B = N \ ((N \ A) ∪ (N \B)). For every property P of natural numbers, we say that P is true ω-almost surely if the set S of numbers with property P satisfies ω(S) = 1. Thus is ω-almost surely every natural number satisfies P and ω-almost surely every natural number satisfies Q, then ω-almost surely every natural number satisfies P and Q. We are going to use this property of ultrafilters several times later without reference. For every sequence of real numbers ri ≥ 0 one can define the limit limω ri as the (unique) number r ∈ R∪ {∞} such that for every ǫ we have |ri − r| < ǫ ω-almost surely for all i ∈ N (see [2]). Choose a sequence of scaling constants dn > 0 such that limω dn = ∞ and define a pseudo-metric dist on the Cartesian power G as dist((ui), (vi)) = lim ω dist(ui, vi) di . Let GNb be the connected component of the sequence 1̄ = (1, 1, . . .) ∈ G . Thus GNb consists of all the elements from G at finite distance from 1̄. Let ∼ be the equivalence relation (ui) ∼ (vi) if and only if dist((ui), (vi)) = 0. Then dist induces a metric dist ω on GNb / ∼, and GNb / ∼ with this metric is called the asymptotic cone of G corresponding to the ultrafilter ω and the sequence of scaling constants (dn), denoted Con (G, (di)) (for more details see [2, 3, 9]). Elements of Con(G, (dn)) corresponding to sequences (yi) ∈ G N will be denoted 2000 Mathematics Subject Classification. Primary 20F65; Secondary 20F69, 20F38, 22F50.