Primitive groups, graph endomorphisms and synchronization

Fixed points of endomorphisms of graph groups

It is shown, for a given graph group G, that the fixed point subgroup Fix' is finitely generated for every endomorphism ' of G if and only if G is a free product of free abelian groups. The same conditions hold for the subgroup of periodic points. Similar results are obtained for automorphisms if the dependence graph of G is a transitive forest.

Endomorphisms of Relatively Hyperbolic Groups

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. • If H is a non-parabolic subgroup of a relatively hyperb...

Endomorphisms and synchronization, 2: Graphs and transformation monoids

On Endomorphisms of Surface Mapping Class Groups

We prove in this paper that every endomorphism of the mapping class group of certain orientable surfaces onto a subgroup of finite index is in fact an automorphism.

Endomorphisms of Compact Abelian Groups Minimize Entropy

The entropy conjecture for tori has been settled a long time ago by Misiurewicz and Przytycki. We show that their result extends from tori to compact and connected abelian groups. We give stronger results for hyperbolic automorphisms. Throughout this paper G denotes a compact and connected metric abelian group. The group operation on G is written additively.