Fixed subgroups of automorphisms of relatively hyperbolic groups

Details Fixed subgroups of automorphisms of relatively hyperbolic groups Fixed subgroups of automorphisms of relatively hyperbolic groups

2010-07-14

arxiv.org/abs/1007.2361v2

Mathematics

Group Theory

Version 2: 18 pages, 4 figures; expanded the Introduction and improved the overall exposition following referee's suggestions

Scientific paper

Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex. It follows that the fixed subgroup is itself relatively hyperbolic with respect to a natural family of peripheral subgroups. If all peripheral subgroups of G are slender (respectively, slender and coherent), our result implies that the fixed subgroup of every automorphism of G is finitely generated (respectively, finitely presented). In particular, this happens when G is a limit group, and thus for any automorphism \phi of G, Fix(\phi) is a limit subgroup of G.

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