Stallings graphs for quasi-convex subgroups of hyperbolic groups Pascal Weil (CNRS, Universit´ e de Bordeaux) Joint work with Olga Kharlampovich (CUNY) and Alexei Miasnikov (Stevens Institute) GAGTA 7, New York City, May 2013 Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Stallings graphs ◮ Stallings graphs have become a standard for representing finitely generated subgroups of free groups and solving algorithmic problems on them Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Stallings graphs ◮ Stallings graphs have become a standard for representing finitely generated subgroups of free groups and solving algorithmic problems on them ◮ They are effectively computable, they help solve efficiently the membership problem, compute intersections, decide finite index, and many other problems. Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Stallings graphs ◮ Stallings graphs have become a standard for representing finitely generated subgroups of free groups and solving algorithmic problems on them ◮ They are effectively computable, they help solve efficiently the membership problem, compute intersections, decide finite index, and many other problems. ◮ Efficient solutions because of automata-theoretic flavor Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Stallings graphs ◮ Stallings graphs have become a standard for representing finitely generated subgroups of free groups and solving algorithmic problems on them ◮ They are effectively computable, they help solve efficiently the membership problem, compute intersections, decide finite index, and many other problems. ◮ Efficient solutions because of automata-theoretic flavor ◮ We would like something similar for finitely generated subgroups of other groups Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Stallings graphs ◮ Stallings graphs have become a standard for representing finitely generated subgroups of free groups and solving algorithmic problems on them ◮ They are effectively computable, they help solve efficiently the membership problem, compute intersections, decide finite index, and many other problems. ◮ Efficient solutions because of automata-theoretic flavor ◮ We would like something similar for finitely generated subgroups of other groups ◮ More precisely, a constructible automaton (labeled graph) canonically associated with each subgroup, solving at least the membership problem. Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Many results already ◮ Not a new idea Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Many results already ◮ Not a new idea ◮ Kapovich, Miasnikov, Weidmann (2005): the membership problem for subgroups of certain graphs of groups Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Many results already ◮ Not a new idea ◮ Kapovich, Miasnikov, Weidmann (2005): the membership problem for subgroups of certain graphs of groups ◮ Markus-Epstein (2007): construct a Stallings graph for the subgroups of amalgamated products of finite groups Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Many results already ◮ Not a new idea ◮ Kapovich, Miasnikov, Weidmann (2005): the membership problem for subgroups of certain graphs of groups ◮ Markus-Epstein (2007): construct a Stallings graph for the subgroups of amalgamated products of finite groups ◮ Silva, Soler-Escriva, Ventura (2011) for subgroups of virtually free groups Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Many results already ◮ Not a new idea ◮ Kapovich, Miasnikov, Weidmann (2005): the membership problem for subgroups of certain graphs of groups ◮ Markus-Epstein (2007): construct a Stallings graph for the subgroups of amalgamated products of finite groups ◮ Silva, Soler-Escriva, Ventura (2011) for subgroups of virtually free groups ◮ In all three cases: rely on a folding process Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Many results already ◮ Not a new idea ◮ Kapovich, Miasnikov, Weidmann (2005): the membership problem for subgroups of certain graphs of groups ◮ Markus-Epstein (2007): construct a Stallings graph for the subgroups of amalgamated products of finite groups ◮ Silva, Soler-Escriva, Ventura (2011) for subgroups of virtually free groups ◮ In all three cases: rely on a folding process ◮ Markus-Epstein and SSV rely on a well-chosen set of representatives Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Assumptions on G and H ◮ Need to impose constraints on G and H ≤ G : in general not even the word problem for G is decidable Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Assumptions on G and H ◮ Need to impose constraints on G and H ≤ G : in general not even the word problem for G is decidable ◮ and even in good situations (e.g. G is automatic, or even hyperbolic), not every finitely generated subgroup admits a regular set of representatives Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Assumptions on G and H ◮ Need to impose constraints on G and H ≤ G : in general not even the word problem for G is decidable ◮ and even in good situations (e.g. G is automatic, or even hyperbolic), not every finitely generated subgroup admits a regular set of representatives ◮ We want G = � A | R � to be automatic (e.g. hyperbolic), Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Assumptions on G and H ◮ Need to impose constraints on G and H ≤ G : in general not even the word problem for G is decidable ◮ and even in good situations (e.g. G is automatic, or even hyperbolic), not every finitely generated subgroup admits a regular set of representatives ◮ We want G = � A | R � to be automatic (e.g. hyperbolic), ◮ and H to be quasi-convex. That is: there exists a constant k such that every geodesic to an element of H stays within the k -neighborhood of H Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
Assumptions on G and H ◮ Need to impose constraints on G and H ≤ G : in general not even the word problem for G is decidable ◮ and even in good situations (e.g. G is automatic, or even hyperbolic), not every finitely generated subgroup admits a regular set of representatives ◮ We want G = � A | R � to be automatic (e.g. hyperbolic), ◮ and H to be quasi-convex. That is: there exists a constant k such that every geodesic to an element of H stays within the k -neighborhood of H ◮ Note that in [Markus-Epstein] or [SSV], we are dealing with locally quasi-convex groups: all finitely generated subgroups are quasi-convex Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
More precisely ◮ Assume G = � A | R � , L is a regular set of representatives, µ : F ( A ) → G (e.g. L = L geod = geodesics, Dehn irreducible words) Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
More precisely ◮ Assume G = � A | R � , L is a regular set of representatives, µ : F ( A ) → G (e.g. L = L geod = geodesics, Dehn irreducible words) ◮ If H ≤ G , µ − 1 ( H ) is a subgroup, not always finitely generated Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
More precisely ◮ Assume G = � A | R � , L is a regular set of representatives, µ : F ( A ) → G (e.g. L = L geod = geodesics, Dehn irreducible words) ◮ If H ≤ G , µ − 1 ( H ) is a subgroup, not always finitely generated ◮ Define: H is L -quasi-convex if L ∩ µ − 1 ( H ) is a regular language Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
More precisely ◮ Assume G = � A | R � , L is a regular set of representatives, µ : F ( A ) → G (e.g. L = L geod = geodesics, Dehn irreducible words) ◮ If H ≤ G , µ − 1 ( H ) is a subgroup, not always finitely generated ◮ Define: H is L -quasi-convex if L ∩ µ − 1 ( H ) is a regular language ◮ H is quasi-convex if it is L geod -quasi-convex Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
General outline of our results ◮ If G is automatic, L is the corresponding regular set of representatives and H ≤ G is L -quasi-convex, we construct effectively a Stallings graph for H Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
General outline of our results ◮ If G is automatic, L is the corresponding regular set of representatives and H ≤ G is L -quasi-convex, we construct effectively a Stallings graph for H ◮ we solve the membership problem Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
General outline of our results ◮ If G is automatic, L is the corresponding regular set of representatives and H ≤ G is L -quasi-convex, we construct effectively a Stallings graph for H ◮ we solve the membership problem ◮ we find the constant of L -quasi-convexity Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
General outline of our results ◮ If G is automatic, L is the corresponding regular set of representatives and H ≤ G is L -quasi-convex, we construct effectively a Stallings graph for H ◮ we solve the membership problem ◮ we find the constant of L -quasi-convexity ◮ we decide finite index Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic
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