10 research outputs found
Complexity volumes of splittable groups
Using graph of groups decompositions of finitely generated groups, we define
Euler characteristic type invariants which are non-zero in many interesting
classes of finitely presented, hyperbolic, limit and CSA groups, including
elementarily free groups and one-ended torsion-free hyperbolic groups whose JSJ
decomposition contains a maximal hanging Fuchsian vertex group.Comment: Major revisions following the suggestions of the referee. To appear
in Journal of Algebr
Finiteness results for subgroups of finite extensions
We discuss in the context of finite extensions two classical theorems of
Takahasi and Howson on subgroups of free groups. We provide bounds for the rank
of the intersection of subgroups within classes of groups such as virtually
free groups, virtually nilpotent groups or fundamental groups of finite graphs
of groups with virtually polycyclic vertex groups and finite edge groups. As an
application of our generalization of Takahasi's Theorem, we provide an uniform
bound for the rank of the periodic subgroup of any endomorphism of the
fundamental group of a given finite graph of groups with finitely generated
virtually nilpotent vertex groups and finite edge groups.Comment: 20 pages; no figures. Keywords: finite extensions, Howson's Theorem,
Hanna Neumann Conjecture, Takahasi's Theorem, periodic subgroup
Fixed points of endomorphisms of graph groups
It is shown, for a given graph group , that the fixed point subgroup
Fix is finitely generated for every endomorphism of if
and only if 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 is a transitive forest.Comment: 9 page