Geodesic
Bounds for fixed points and fixed subgroups on surfaces and graphs
Jiang, Boju1 ; Wang, Shida2 ; Zhang, Qiang3
1Department of Mathematics, Peking University, Beijing 100871, China
2Department of Mathematics, Indiana University, Bloomington IN 47405, USA
3School of Science, Xi’an Jiaotong University, Xi’an 710049, China
Algebraic and Geometric Topology, Tome 11 (2011) no. 4, pp. 2297-2318
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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.

DOI : 10.2140/agt.2011.11.2297
Keywords: fixed point class, index, fixed subgroup, rank, surface map, surface group endomorphism, graph map, free group endomorphism

Jiang, Boju  1   ; Wang, Shida  2   ; Zhang, Qiang  3

1 Department of Mathematics, Peking University, Beijing 100871, China
2 Department of Mathematics, Indiana University, Bloomington IN 47405, USA
3 School of Science, Xi’an Jiaotong University, Xi’an 710049, China 
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Jiang, Boju; Wang, Shida; Zhang, Qiang. Bounds for fixed points and fixed subgroups on surfaces and graphs. Algebraic and Geometric Topology, Tome 11 (2011) no. 4, pp. 2297-2318. doi: 10.2140/agt.2011.11.2297

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