Finitely generated groups are built that satisfy Burnside laws with limit probability 1 (or 0, or any partial limit in [0,1]) under ball measures and random walks, resolving open questions on sensitivity to generators and coexistence of laws.
Degree of commutativity of infinite groups
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abstract
We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.
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Asymptotic Burnside laws
Finitely generated groups are built that satisfy Burnside laws with limit probability 1 (or 0, or any partial limit in [0,1]) under ball measures and random walks, resolving open questions on sensitivity to generators and coexistence of laws.