Algebraic & Geometric Topology 11 (2011) 2297–2318

DOI: 10.2140/agt.2011.11.2297

Abstract

We consider selfmaps of hyperbolic surfaces and graphs, and give some bounds involving the rank and the index of fixed point classes. One consequence is a rank bound for fixed subgroups of surface group endomorphisms, similar to the Bestvina–Handel bound (originally known as the Scott conjecture) for free group automorphisms.

When the selfmap is homotopic to a homeomorphism, we rely on Thurston’s classification of surface automorphisms. When the surface has boundary, we work with its spine, and Bestvina–Handel’s theory of train track maps on graphs plays an essential role.

It turns out that the control of empty fixed point classes (for surface automorphisms) presents a special challenge. For this purpose, an alternative definition of fixed point class is introduced, which avoids covering spaces hence is more convenient for geometric discussions.

Keywords

fixed point class, index, fixed subgroup, rank, surface map, surface group endomorphism, graph map, free group endomorphism

Mathematical Subject Classification

Primary: 55M20, 57M07

Secondary: 20F34, 57M15, 57N05

References

Publication

Received: 10 October 2010
Revised: 16 February 2011
Accepted: 21 February 2011
Published: 26 August 2011

Authors