Algebraic & Geometric Topology

Bounds for fixed points and fixed subgroups on surfaces and graphs

Boju Jiang, Shida Wang, and Qiang Zhang

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715270

Digital Object Identifier
doi:10.2140/agt.2011.11.2297

Mathematical Reviews number (MathSciNet)
MR2826940

Zentralblatt MATH identifier
1232.55006

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.agt/1513715270


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References

  • M Bestvina, M Handel, Train tracks and automorphisms of free groups, Ann. of Math. $(2)$ 135 (1992) 1–51
  • W Dicks, E Ventura, The group fixed by a family of injective endomorphisms of a free group, Contemporary Mathematics 195, American Mathematical Society, Providence, RI (1996)
  • A Fathi, F Laudenbach, V Poénaru, editors, Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France, Paris (1979) With an English summary
  • B J Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics 14, American Mathematical Society, Providence, R.I. (1983)
  • B Jiang, Bounds for fixed points on surfaces, Math. Ann. 311 (1998) 467–479
  • B J Jiang, J H Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993) 67–89
  • R C Lyndon, P E Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer, Berlin (1977)
  • H Masur, J Smillie, Quadratic differentials with prescribed singularities and pseudo–Anosov diffeomorphisms, Comment. Math. Helv. 68 (1993) 289–307
  • W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. $($N.S.$)$ 19 (1988) 417–431
  • H Zieschang, Über einfache Kurven auf Vollbrezeln, Abh. Math. Sem. Univ. Hamburg 25 (1961/1962) 231–250