Illinois Journal of Mathematics

Exponentially generic subsets of groups

Robert Gilman, Alexei Miasnikov, and Denis Osin

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In this paper, we study the generic, i.e., typical, behavior of finitely generated subgroups of hyperbolic groups and also the generic behavior of the word problem for amenable groups. We show that a random set of elements of a nonelementary word hyperbolic group is very likely to be a set of free generators for a nicely embedded free subgroup. We also exhibit some finitely presented amenable groups for which the restriction of the word problem is unsolvable on every sufficiently large subset of words.

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Illinois J. Math. Volume 54, Number 1 (2010), 371-388.

First available in Project Euclid: 9 March 2011

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Zentralblatt MATH identifier

Primary: 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70]
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 43A07: Means on groups, semigroups, etc.; amenable groups


Gilman, Robert; Miasnikov, Alexei; Osin, Denis. Exponentially generic subsets of groups. Illinois J. Math. 54 (2010), no. 1, 371--388.

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