## Conjugacy languages in groups

### Laura Ciobanu, Susan Hermiller, Derek Holt, and Sarah Rees

#### Keywords

conjugacy problem, geodesics, formal languages

#### Status

Submitted for publication.

#### Abstract

We study the regularity of several languages derived from conjugacy classes in a finitely generated group $G$ for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the \wsphcl \ in virtually cyclic groups, proving one direction of a conjecture of Rivin. \bigskip \noindent 2010 Mathematics Subject Classification: 20F65, 20E45; 20F67, 20F36. \noindent Key words: Conjugacy growth, regular languages, word hyperbolic groups, Artin groups, Garside groups. \

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