In the 1980's Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups.
In this paper we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for ``sewing'' on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.
Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. It turns out that Stallings' methods can be effectively generalized for the class of amalgams of finite groups. In the present paper we employ subgroup graphs constructed by the generalized Stallings' folding algorithm to solve various algorithmic problems in amalgams of finite groups.
We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stalling's folding process in time O(N log^*(N)), where N is the sum of the word lengths of the given generators of H.
We re-cast in a more combinatorial and computational form the foldings approach of John Stallings and pursue a detailed study of the subgroup structure of free groups. In particular, we introduce the notions of an "algebraic" and a "free" extension for subgroups of free groups and apply these notions to obtain a number of algebraic and algorithmic results.
Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ the generalized Stallings' methods, developed by the author, to solve various algorithmic problems concerning finitely generated subgroups of amalgams of finite groups.
The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. We generalize the pro-$p$ version of it proved by Weigel and the second author to splittings over infinite pro-$p$ groups. This generalization does not have any abstract analogs. We also prove that generalized accessibility of finitely generated pro-$p$ groups is closed for commensurability.
We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers.
Stallings folding theory is modified, using double coset representatives, and to applied to the study of subgroups of amalgamated products of finite rank free groups. As a first application the subgroup membership problem for such groups is shown to be decidable. An algorithm for this problem is constructed and its complexity is analysed. Groups in this class are also shown to possess the Howson property.
In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fund…
We introduce a combinatorial version of Stallings-Bestvina-Feighn-Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups.
The Stallings construction for finitely generated subgroups of free groups is generalized by introducing the concept of Stallings section, which allows an eficient computation of the core of a Schreier graph based on edge folding. It is proved that those groups admitting Stallings sections are precisely finitely generated virtually free groups, through a constructive approach based on Bass-Serre theory. Complexity issues and applications are also discussed.
In this paper, we give some necessary and sufficient conditions for a normal subgroup of an amalgamated product of groups to be finitely generated. We apply these conditions together with Stallings' fibering theorem to prove that an irreducible multilink in a homology 3-sphere fibers if and only if each of its multilink splice components fibers.
Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used by many authors to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ our generalized Stallings' folding method to introduce a procedure, which given a subgroup H of a free product of finite groups reads off its Kurosh decomposition from the subgroup graph of H.
We describe an algorithm that uses Stallings' folding technique to decompose an element of $Aut(F_n)$ as a product of Whitehead automorphisms (and hence as a product of Nielsen transformations.) We use this to give an alternative method of finding a finite generating set for the subgroup of $Aut(F_n)$ that fixes a subset $Y$ of the basis elements, and the subgroup that fixes each element of $Y$ up to conjugacy. We show that the intersection of this latter subgroup with $IA_n$ is also finitely generated.
We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2-dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are a…
Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregroups and in particular to free products with amalgamation, HNN-extensions and virtually free groups, to yield simple and intuitive algorithms and proofs of conjugacy criteria.
In 1971 J. Stallings introduced a generalisation of amalgamated products of groups -- called a pregroup, which is a particular kind of a partial group. He defined the universal group U(P) of a pregroup P to be a universal object (in the sense of category theory) extending the partial operations on P to group operations on U(P). This turns out to be a versatile and convenient generalisation of classical group constructions: HNN-extensions and amalgamated products. In this respect the following general question arises. Which properties of pregroups, or relations between pregroups, carry over to …
Triangles of groups have been introduced by Gersten and Stallings. They are, roughly speaking, a generalisation of the amalgamated free product of two groups and occur in the framework of Corson diagrams. First, we prove an intersection theorem for Corson diagrams. Then, we focus on triangles of groups. It has been shown by Howie and Kopteva that the colimit of a hyperbolic triangle of groups contains a non-abelian free subgroup. We give two natural conditions, each of which ensures that the colimit of a non-spherical triangle of groups either contains a non-abelian free subgroup or is virtual…
Suppose that $F$ is a free group and $k$ is a natural number. We show that the fully compressed membership problem for $k$-generated subgroups of $F$ is solvable in polynomial time. In order to do this, we adapt the theory of Stallings' foldings to handle edges with compressed labels. This partially answers a question of Markus Lohrey.
Let $F$ be any finite-rank free group, and $R$ be any finite subset of $\{g, [g]: g \in F-\{1\}\}$, where $[g]:= \{fgf^{-1}:f\in F\}$. By an $R$-allocating $F$-factorization we mean a set $\mathcal{H}$ of nontrivial subgroups of $F$ such that $\ast_{H \in \mathcal{H}} H = F$ and $R \subseteq \{h, [h] : h \in H, H\in \mathcal{H}\}$. We show that Whitehead's (fast) cutvertex algorithm inputs the pair $(F,R)$ and outputs a maximum-size $R$-allocating $F$-factorization. Richard Stong showed this in the case where $R \subseteq F$ or $R \subseteq \{[g] : g \in F\}$, thereby unifying and generalizing…
We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has …
