Fixed subgroups of automorphisms of relatively hyperbolic groups
Fixed subgroups of automorphisms of relatively hyperbolic groups
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.
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Osin, Denis
32a9932c-f439-4b83-b639-1a53ac6bf6f5
September 2012
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Osin, Denis
32a9932c-f439-4b83-b639-1a53ac6bf6f5
Minasyan, Ashot and Osin, Denis
(2012)
Fixed subgroups of automorphisms of relatively hyperbolic groups.
The Quarterly Journal of Mathematics, 63 (3).
(doi:10.1093/qmath/har015).
Abstract
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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Submitted date: 10 September 2010
Published date: September 2012
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Local EPrints ID: 181801
URI: http://eprints.soton.ac.uk/id/eprint/181801
ISSN: 0033-5606
PURE UUID: 9a47adc8-f260-431b-8ff5-fc98ace7d0da
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Date deposited: 27 Apr 2011 09:12
Last modified: 15 Mar 2024 03:29
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Denis Osin
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