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

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.Comment: 11 pages, 1 figure. Section 3 is replaced by Theorem 1.3. More detailed arguments. Added new references. To appear in Journal of Pure and Applied 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 $G$, that the fixed point subgroup Fix$\,\varphi$ is finitely generated for every endomorphism $\varphi$ 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.Comment: 9 page

Keap1/Nrf2 Signaling Pathway.

Nuclear factor, erythroid 2-like transcription factor 2 (Nrf2) and its cytoplasmic inhibitor, kelch-like ECH-associated protein 1 (Keap1), comprise a redox-responsive endogenous antioxidant defense module that orchestrates the expression of cytoprotective genes to maintain homeostasis [...]