Actions of finite groups on graphs and related automorphisms of free groups

Determining whether a finite group of outer automorphisms of a free group is geometric

“…However, since the non-trivialĝ 3 fixes all three edges of star(v), it is clear thatĜ is not geometric by Theorem 3.6. Since all of the edges of are in one orbit, the work of Krstić [8] proves that this is the only reduced group of graph automorphisms which realizes O(Ĝ). Thus O(Ĝ) is not geometric and the conditions of Theorem 3.10 are not sufficient for non-abelian finite groups of outer automorphisms.…”

“…Given such a graph and homeomorphisms, Krstić [8] in 1989 showed how to generate all the finitely many reduced graphs (no invariant forests) that realize G. From here, one can check certain properties at each of the vertices and edges of each graph, using the work of Los and Nitecki [9], to determine if G is geometric. In this paper, we follow these steps to obtain necessary and sufficient conditions for G to be geometric, based on the finite groups of automorphisms which project into G and the fixed subgroups of those automorphisms.…”

Context-Free Groups and Their Structure Trees

“…The problem is to find the minimal cuts, i.e., to decide whether a given cut is minimal with respect to some bi-infinite simple path. If this could be done in elementary time for graphs of finite tree width, it would lead to an elementary time algorithm for the isomorphism problem of contextfree groups by first constructing the graphs of groups and then using Krstic's algorithm ( [20]) to check whether the fundamental groups are isomorphic.…”

On the mod-p cohomology of Out(F2(p−1Glover1, Henn2 2010Journal of Pure and Applied Algebra1000View full textAdd to dashboardCite

“…This approach has also been used by Jensen [6] in the case of Aut(F 2(p−1) ). Our analysis of the normalizers of elementary abelian subgroups of Out(F 2(p−1) ) is quite similar to that of the normalizers in Aut(F 2(p−1) ) which was carried out in [6] using results of Krstic [10].…”