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2011 Bounds for fixed points and fixed subgroups on surfaces and graphs
Boju Jiang, Shida Wang, Qiang Zhang
Algebr. Geom. Topol. 11(4): 2297-2318 (2011). DOI: 10.2140/agt.2011.11.2297


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.


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Boju Jiang. Shida Wang. Qiang Zhang. "Bounds for fixed points and fixed subgroups on surfaces and graphs." Algebr. Geom. Topol. 11 (4) 2297 - 2318, 2011.


Received: 10 October 2010; Revised: 16 February 2011; Accepted: 21 February 2011; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1232.55006
MathSciNet: MR2826940
Digital Object Identifier: 10.2140/agt.2011.11.2297

Primary: ‎55M20 , 57M07
Secondary: 20F34 , 57M15 , 57N05

Keywords: fixed point class , fixed subgroup , free group endomorphism , graph map , Index , ‎rank‎ , surface group endomorphism , surface map

Rights: Copyright © 2011 Mathematical Sciences Publishers


Vol.11 • No. 4 • 2011
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