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Bounds for fixed points
and fixed subgroups on surfaces and graphs
Boju Jiang, Shida Wang and Qiang Zhang |
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Algebraic & Geometric Topology 11 (2011) 2297–2318
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Abstract |
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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. |
Keywords
fixed point class, index, fixed subgroup, rank, surface
map, surface group endomorphism, graph map, free group
endomorphism
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Mathematical Subject Classification 2000
Primary: 55M20, 57M07
Secondary: 20F34, 57M15, 57N05
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References |
Publication
Received: 10 October 2010
Revised: 16 February 2011
Accepted: 21 February 2011
Published: 26 August 2011
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Authors |
| Boju Jiang | |
| Department of Mathematics Peking University Beijing 100871 China |
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| Shida Wang | |
| Department of Mathematics Indiana University Bloomington IN 47405 USA |
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| Qiang Zhang | |
| School of Science Xi’an Jiaotong University Xi’an 710049 China |
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