## Illinois Journal of Mathematics

### Exponentially generic subsets of groups

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 9 March 2011

Permanent link to this document
http://projecteuclid.org/euclid.ijm/1299679753

Zentralblatt MATH identifier
05882261

Mathematical Reviews number (MathSciNet)
MR2777000

#### Citation

Gilman, Robert; Miasnikov, Alexei; Osin, Denis. Exponentially generic subsets of groups. Illinois Journal of Mathematics 54 (2010), no. 1, 371--388. http://projecteuclid.org/euclid.ijm/1299679753.

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