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