Bounds for fixed points and fixed subgroups on surfaces and graphs
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.
scientific article; zbMATH DE number 5959841