Bounds for fixed points and fixed subgroups on surfaces and graphs
type: Item
instance of: scholarly article
title: Bounds for fixed points and fixed subgroups on surfaces and graphs
zbMATH Open document ID: 1232.55006
author: Shida Wang
author: Qiang Zhang
author: Boju Jiang
published in: Algebraic \& Geometric Topology
publication date: 2011-10-18T00:00:00Z
review text: A crucial invariant in Nielsen fixed point theory is the so-called Nielsen number, which is defined to be the number of fixed point classes with non-zero indices. The sum of indices of all fixed point classes of a given selfmap is just the Lefschetz number. It is natural to ask for more relations between fixed point indices and other topological invariants. The paper under review contains some very interesting results in this direction: bounds (from below or above) for indices of fixed point classes of selfmaps on hyperbolic surfaces or graphs. The invariants involved are the Euler characteristic of the underlying spaces and fixed subgroups of the induced endomorphisms on the fundamental group based on a point in a given fixed point class. The main results in this paper imply \textit{M. Bestvina} and \textit{M. Handel}'s rank inequality [Ann. Math. (2) 135, No. 1, 1--51 (1992; Zbl 0757.57004)] which solved the Scott conjecture for free group automorphisms, and also imply a result of first author [Math. Ann. 311, No.3, 467--479 (1998; Zbl 0903.55003)] about bounds for fixed points on surfaces. Both the results and the methods in this paper illustrate some non-trivial interactions between Nielsen fixed point theory and group theory.
reviewed by: Xuezhi Zhao
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 5959841
zbMATH Keywords: fixed point class
zbMATH Keywords: index
zbMATH Keywords: fixed subgroup
zbMATH Keywords: rank
zbMATH Keywords: surface map
zbMATH Keywords: graph map
zbMATH Keywords: free group
MaRDI profile type: Publication
full work available at URL: agt.2011.11.2297
OpenAlex ID: W2002680896
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DOI: 10.2140/AGT.2011.11.2297
description: scientific article; zbMATH DE number 5959841