Algebraic & Geometric Topology

Bounds for fixed points and fixed subgroups on surfaces and graphs

Boju Jiang, Shida Wang, and Qiang Zhang

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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.

Article information

Algebr. Geom. Topol., Volume 11, Number 4 (2011), 2297-2318.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M20: Fixed points and coincidences [See also 54H25] 57M07: Topological methods in group theory
Secondary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 57M15: Relations with graph theory [See also 05Cxx] 57N05: Topology of $E^2$ , 2-manifolds

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


Jiang, Boju; Wang, Shida; Zhang, Qiang. Bounds for fixed points and fixed subgroups on surfaces and graphs. Algebr. Geom. Topol. 11 (2011), no. 4, 2297--2318. doi:10.2140/agt.2011.11.2297.

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