Finite automata for Schreier graphs of virtually free groups

Abstract

The Stallings construction for f.g. subgroups of free groups is generalized by introducing the concept of Stallings section, which allows efficient computation of the core of a Schreier graph based on edge folding. It is proved that the groups that admit Stallings sections are precisely the f.g. virtually free groups, this is proved through a constructive approach based on Bass-Serre theory. Complexity issues and applications are also discussed.Peer ReviewedPostprint (published version

On the generalized membership problem in relatively hyperbolic groups

“…Markus-Epstein [25] constructs a Stallings graph for the subgroups of amalgamated products of finite groups. Silva, Soler-Escriva, Ventura [34] do the same for subgroups of virtually free groups. Here again, the groups considered are locally quasi-convex, and the authors rely on a folding process, much like in the free group case, and a well-chosen set of representatives.…”

On the transition monoid of the Stallings automaton of a subgroup of a free group

“…Finally, one could pursue these same goals regarding structures that generalize Stallings automata for some wider classes of groups [11,15,20,12].…”

Statistics of subgroups of the modular group

“…It is well-known that PSL 2 (Z) is hyperbolic, virtually free and that every one of its finitely generated subgroup is quasi-convex. It follows that every finitely generated subgroup H of PSL 2 (Z) is uniquely represented by a finite, labeled, rooted graph (Γ(H), v), called its Stallings graph [14] (see also [21,35]), which can be efficiently computed, see below. The definition of the Stallings graph of a subgroup H is as follows: we first consider the graph Schreier(PSL 2 (Z), H), whose vertices are the cosets Hg (g ∈ PSL 2 (Z)), with an a-labeled edge from Hg to Hga and a b-labeled edge from Hg to Hgb for every g ∈ PSL 2 (Z).…”

“…Remark A.2 Using Equations ( 34) and (35), one can compute the values of t 2 (m, k) and t 3 (m, k) for all m ≤ n and all k ≤ ℓ in O(nℓ) time in the unit cost model.…”