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Bounds for fixed points and fixed subgroups on surfaces and graphs

Boju Jiang, Shida Wang and Qiang Zhang

Algebraic & Geometric Topology 11 (2011) 2297–2318

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.

fixed point class, index, fixed subgroup, rank, surface map, surface group endomorphism, graph map, free group endomorphism
Mathematical Subject Classification 2000
Primary: 55M20, 57M07
Secondary: 20F34, 57M15, 57N05
Received: 10 October 2010
Revised: 16 February 2011
Accepted: 21 February 2011
Published: 26 August 2011
Boju Jiang
Department of Mathematics
Peking University
Beijing 100871
Shida Wang
Department of Mathematics
Indiana University
Bloomington IN 47405
Qiang Zhang
School of Science
Xi’an Jiaotong University
Xi’an 710049