401 research outputs found
On the lattice of subgroups of a free group: complements and rank
A -complement of a subgroup is a subgroup such that . If we also ask
to have trivial intersection with , then we say that is a
-complement of . The minimum possible rank of a -complement
(resp. -complement) of is called the -corank (resp.
-corank) of . We use Stallings automata to study these notions and
the relations between them. In particular, we characterize when complements
exist, compute the -corank, and provide language-theoretical descriptions
of the sets of cyclic complements. Finally, we prove that the two notions of
corank coincide on subgroups that admit cyclic complements of both kinds.Comment: 27 pages, 5 figure
Stallings graphs for quasi-convex subgroups
We show that one can define and effectively compute Stallings graphs for
quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or
right-angled Artin groups). These Stallings graphs are finite labeled graphs,
which are canonically associated with the corresponding subgroups. We show that
this notion of Stallings graphs allows a unified approach to many algorithmic
problems: some which had already been solved like the generalized membership
problem or the computation of a quasi-convexity constant (Kapovich, 1996); and
others such as the computation of intersections, the conjugacy or the almost
malnormality problems.
Our results extend earlier algorithmic results for the more restricted class
of virtually free groups. We also extend our construction to relatively
quasi-convex subgroups of relatively hyperbolic groups, under certain
additional conditions.Comment: 40 pages. New and improved versio
A fast algorithm for Stallings foldings over virtually free groups
We give a simple algorithm to solve the subgroup membership problem for
virtually free groups. For a fixed virtually free group with a fixed generating
set , the subgroup membership problem is uniformly solvable in time
where is the sum of the word lengths of the inputs with
respect to . For practical purposes, this can be considered to be linear
time. The algorithm itself is simple and concrete examples are given to show
how it can be used for computations in and
. We also give an algorithm to decide whether a
finitely generated subgroup is isomorphic to a free group.Comment: 26 pages, 9 Figure
- …