Stallings graphs for quasi-convex subgroups of hyperbolic groups - - PowerPoint PPT Presentation

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 = Lgeod = 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 = Lgeod = 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 = Lgeod = 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 = Lgeod = 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 Lgeod-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

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 ◮ we compute finite intersections,

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

The case of hyperbolic groups

◮ If G is hyperbolic, we can find an automatic structure for

which the set of representatives is Lgeod

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

The case of hyperbolic groups

◮ If G is hyperbolic, we can find an automatic structure for

which the set of representatives is Lgeod

◮ Then L-quasi-convexity is quasi-convexity

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

The case of hyperbolic groups

◮ If G is hyperbolic, we can find an automatic structure for

which the set of representatives is Lgeod

◮ Then L-quasi-convexity is quasi-convexity ◮ So we can decide membership and finite index, compute finite

intersections for quasi-convex subgroups of a hyperbolic group

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

The case of hyperbolic groups

◮ If G is hyperbolic, we can find an automatic structure for

which the set of representatives is Lgeod

◮ Then L-quasi-convexity is quasi-convexity ◮ So we can decide membership and finite index, compute finite

intersections for quasi-convex subgroups of a hyperbolic group

◮ These are not new results, but our construction provides a

unified tool – which surely can be used for other decision problems

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Definition of a Stallings graph!

◮ Schreier graph of H: vertex set the Hg (g ∈ G); draw an

a-labeled edge (a ∈ A) from Hg to Hgµ(a)

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Definition of a Stallings graph!

◮ Schreier graph of H: vertex set the Hg (g ∈ G); draw an

a-labeled edge (a ∈ A) from Hg to Hgµ(a)

◮ w labels a loop at vertex H if and only if µ(w) ∈ H

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Definition of a Stallings graph!

◮ Schreier graph of H: vertex set the Hg (g ∈ G); draw an

a-labeled edge (a ∈ A) from Hg to Hgµ(a)

◮ w labels a loop at vertex H if and only if µ(w) ∈ H ◮ Stallings graph for H with respect to L: the fragment ΓL(H)

• f the Schreier graph, spanned by the loops labeled by the

L-representatives of the elements of H

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Definition of a Stallings graph!

◮ Schreier graph of H: vertex set the Hg (g ∈ G); draw an

a-labeled edge (a ∈ A) from Hg to Hgµ(a)

◮ w labels a loop at vertex H if and only if µ(w) ∈ H ◮ Stallings graph for H with respect to L: the fragment ΓL(H)

• f the Schreier graph, spanned by the loops labeled by the

L-representatives of the elements of H

◮ This generalizes the free group case; this is uniquely

associated with H (and L)

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Definition of a Stallings graph!

◮ Schreier graph of H: vertex set the Hg (g ∈ G); draw an

a-labeled edge (a ∈ A) from Hg to Hgµ(a)

◮ w labels a loop at vertex H if and only if µ(w) ∈ H ◮ Stallings graph for H with respect to L: the fragment ΓL(H)

• f the Schreier graph, spanned by the loops labeled by the

L-representatives of the elements of H

◮ This generalizes the free group case; this is uniquely

associated with H (and L)

◮ It is with this definition in mind that we proceed with the

construction

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

First, a completion process

◮ Given h1, . . . hk ∈ F(A) such that H = µ(h1), . . . , µ(hk): let

H0 = h1, · · · , hk ≤ F(A) and (Γ0, 1) be its Stallings graph

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

First, a completion process

◮ Given h1, . . . hk ∈ F(A) such that H = µ(h1), . . . , µ(hk): let

H0 = h1, · · · , hk ≤ F(A) and (Γ0, 1) be its Stallings graph

◮ (Γi+1, 1) is obtained from (Γi, 1) by gluing every relator at

every vertex and then folding

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

First, a completion process

◮ Given h1, . . . hk ∈ F(A) such that H = µ(h1), . . . , µ(hk): let

H0 = h1, · · · , hk ≤ F(A) and (Γ0, 1) be its Stallings graph

◮ (Γi+1, 1) is obtained from (Γi, 1) by gluing every relator at

every vertex and then folding

◮ a sequence of reduced rooted graphs; usually infinite

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

First, a completion process

◮ Given h1, . . . hk ∈ F(A) such that H = µ(h1), . . . , µ(hk): let

H0 = h1, · · · , hk ≤ F(A) and (Γ0, 1) be its Stallings graph

◮ (Γi+1, 1) is obtained from (Γi, 1) by gluing every relator at

every vertex and then folding

◮ a sequence of reduced rooted graphs; usually infinite ◮ The loops at vertex 1 are always included in µ−1(H),

gradually include all of it

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

First, a completion process

◮ Given h1, . . . hk ∈ F(A) such that H = µ(h1), . . . , µ(hk): let

H0 = h1, · · · , hk ≤ F(A) and (Γ0, 1) be its Stallings graph

◮ (Γi+1, 1) is obtained from (Γi, 1) by gluing every relator at

every vertex and then folding

◮ a sequence of reduced rooted graphs; usually infinite ◮ The loops at vertex 1 are always included in µ−1(H),

gradually include all of it

◮ By L-quasi-convexity: for i large enough, these loops include

L ∩ µ−1(H)

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

First, a completion process

◮ Given h1, . . . hk ∈ F(A) such that H = µ(h1), . . . , µ(hk): let

H0 = h1, · · · , hk ≤ F(A) and (Γ0, 1) be its Stallings graph

◮ (Γi+1, 1) is obtained from (Γi, 1) by gluing every relator at

every vertex and then folding

◮ a sequence of reduced rooted graphs; usually infinite ◮ The loops at vertex 1 are always included in µ−1(H),

gradually include all of it

◮ By L-quasi-convexity: for i large enough, these loops include

L ∩ µ−1(H)

◮ These two properties define a Stallings-like graph for H wrt L

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

First, a completion process

◮ Given h1, . . . hk ∈ F(A) such that H = µ(h1), . . . , µ(hk): let

H0 = h1, · · · , hk ≤ F(A) and (Γ0, 1) be its Stallings graph

◮ (Γi+1, 1) is obtained from (Γi, 1) by gluing every relator at

every vertex and then folding

◮ a sequence of reduced rooted graphs; usually infinite ◮ The loops at vertex 1 are always included in µ−1(H),

gradually include all of it

◮ By L-quasi-convexity: for i large enough, these loops include

L ∩ µ−1(H)

◮ These two properties define a Stallings-like graph for H wrt L ◮ So we eventually construct a Stallings-like graph for H.

• But. . . when to stop?

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Constructing a Stallings-like graph for H wrt L

◮ To decide when to stop: decide whether a reduced rooted

graph (Γ, 1) is Stallings-like for H wrt L. Decide, for each reduced word w labeling a loop at 1, whether the L-representatives of µ(whi) also label loops.

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Constructing a Stallings-like graph for H wrt L

◮ To decide when to stop: decide whether a reduced rooted

graph (Γ, 1) is Stallings-like for H wrt L. Decide, for each reduced word w labeling a loop at 1, whether the L-representatives of µ(whi) also label loops.

◮ This is done using the automatic structure of G

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Constructing a Stallings-like graph for H wrt L

◮ To decide when to stop: decide whether a reduced rooted

graph (Γ, 1) is Stallings-like for H wrt L. Decide, for each reduced word w labeling a loop at 1, whether the L-representatives of µ(whi) also label loops.

◮ This is done using the automatic structure of G ◮ Now we have constructed a Stallings-like graph (Γ, 1)

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Constructing a Stallings-like graph for H wrt L

◮ To decide when to stop: decide whether a reduced rooted

graph (Γ, 1) is Stallings-like for H wrt L. Decide, for each reduced word w labeling a loop at 1, whether the L-representatives of µ(whi) also label loops.

◮ This is done using the automatic structure of G ◮ Now we have constructed a Stallings-like graph (Γ, 1) ◮ And we can solve the membership problem: given w, find an

L-representative, decide whether it labels a loop at 1 in Γ

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Constructing a Stallings-like graph for H wrt L

◮ To decide when to stop: decide whether a reduced rooted

graph (Γ, 1) is Stallings-like for H wrt L. Decide, for each reduced word w labeling a loop at 1, whether the L-representatives of µ(whi) also label loops.

◮ This is done using the automatic structure of G ◮ Now we have constructed a Stallings-like graph (Γ, 1) ◮ And we can solve the membership problem: given w, find an

L-representative, decide whether it labels a loop at 1 in Γ

◮ Note that there is no reason why (Γ, 1) should be embedded

in the Schreier graph

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Finally: construct the Stallings graph of H wrt L

◮ We use our Stallings-like graph (Γ, 1) and the solution of the

membership problem it provides, to compute ΓL(H)

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Finally: construct the Stallings graph of H wrt L

◮ We use our Stallings-like graph (Γ, 1) and the solution of the

membership problem it provides, to compute ΓL(H)

◮ Map Γ to the Schreier graph: map vertex 1 to vertex H. If up

labels a path in Γ from 1 to vertex p, map p to Hµ(up). Decide for all (p, q) whether µ(upu−1

q ) ∈ H

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Finally: construct the Stallings graph of H wrt L

◮ We use our Stallings-like graph (Γ, 1) and the solution of the

membership problem it provides, to compute ΓL(H)

◮ Map Γ to the Schreier graph: map vertex 1 to vertex H. If up

labels a path in Γ from 1 to vertex p, map p to Hµ(up). Decide for all (p, q) whether µ(upu−1

q ) ∈ H ◮ Now we have constructed a subgraph of the Schreier graph

which contains ΓL(H), and which is Stallings-like

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Finally: construct the Stallings graph of H wrt L

◮ We use our Stallings-like graph (Γ, 1) and the solution of the

membership problem it provides, to compute ΓL(H)

◮ Map Γ to the Schreier graph: map vertex 1 to vertex H. If up

labels a path in Γ from 1 to vertex p, map p to Hµ(up). Decide for all (p, q) whether µ(upu−1

q ) ∈ H ◮ Now we have constructed a subgraph of the Schreier graph

which contains ΓL(H), and which is Stallings-like

◮ Since (ΓL, H) is the least rooted subgraph of the Schreier

graph which is Stallings-like: we verify for each vertex whether removing it still yields a Stallings-like graph

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Complexity issues

◮ Finding a Stallings-like graph gives us an L-quasi-convexity

constant for H; computing ΓL(H) gives us the least

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Complexity issues

◮ Finding a Stallings-like graph gives us an L-quasi-convexity

constant for H; computing ΓL(H) gives us the least

◮ But the time needed to do that is not bounded by any

computable function of the size of the input (n = sum of the lengths of the generators)

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Complexity issues

◮ Finding a Stallings-like graph gives us an L-quasi-convexity

constant for H; computing ΓL(H) gives us the least

◮ But the time needed to do that is not bounded by any

computable function of the size of the input (n = sum of the lengths of the generators)

◮ [Otherwise we could decide whether a given tuple of elements

generates a quasi-convex subgroup; and this problem is undecidable]

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Complexity issues

◮ Finding a Stallings-like graph gives us an L-quasi-convexity

constant for H; computing ΓL(H) gives us the least

◮ But the time needed to do that is not bounded by any

computable function of the size of the input (n = sum of the lengths of the generators)

◮ [Otherwise we could decide whether a given tuple of elements

generates a quasi-convex subgroup; and this problem is undecidable]

◮ However, if H has L-quasi-convexity constant a given constant

k, then the number of steps in the completion process is bounded above by a polynomial in n (again: the group G is fixed), see [Brady, Riley, Short]

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Complexity issues

◮ Finding a Stallings-like graph gives us an L-quasi-convexity

constant for H; computing ΓL(H) gives us the least

◮ But the time needed to do that is not bounded by any

computable function of the size of the input (n = sum of the lengths of the generators)

◮ [Otherwise we could decide whether a given tuple of elements

generates a quasi-convex subgroup; and this problem is undecidable]

◮ However, if H has L-quasi-convexity constant a given constant

k, then the number of steps in the completion process is bounded above by a polynomial in n (again: the group G is fixed), see [Brady, Riley, Short]

◮ So, if k is known, then computing ΓL(H) is done in

polynomial time in the total length of the generators

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Applications

◮ Computing the intersection of two quasi-convex subgroups

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Applications

◮ Computing the intersection of two quasi-convex subgroups ◮ Deciding finite index: for this we use an extra condition on

the set L of representatives, namely. . .

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Applications

◮ Computing the intersection of two quasi-convex subgroups ◮ Deciding finite index: for this we use an extra condition on

the set L of representatives, namely. . .

◮ we assume that, for every u ∈ L, there exists an infinite

sequence (vn)n such that for every n, uvn ∈ L and u is a prefix

• f an L-representative of uvnv−1

m uinv for almost all m

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Applications

◮ Computing the intersection of two quasi-convex subgroups ◮ Deciding finite index: for this we use an extra condition on

the set L of representatives, namely. . .

◮ we assume that, for every u ∈ L, there exists an infinite

sequence (vn)n such that for every n, uvn ∈ L and u is a prefix

• f an L-representative of uvnv−1

m uinv for almost all m ◮ Then H has finite index if and only if every word of L can be

read in ΓL(H) starting from the base vertex

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Applications

◮ Computing the intersection of two quasi-convex subgroups ◮ Deciding finite index: for this we use an extra condition on

the set L of representatives, namely. . .

◮ we assume that, for every u ∈ L, there exists an infinite

sequence (vn)n such that for every n, uvn ∈ L and u is a prefix

• f an L-representative of uvnv−1

m uinv for almost all m ◮ Then H has finite index if and only if every word of L can be

read in ΓL(H) starting from the base vertex

◮ This is decidable

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Applications

◮ Computing the intersection of two quasi-convex subgroups ◮ Deciding finite index: for this we use an extra condition on

the set L of representatives, namely. . .

◮ we assume that, for every u ∈ L, there exists an infinite

sequence (vn)n such that for every n, uvn ∈ L and u is a prefix

• f an L-representative of uvnv−1

m uinv for almost all m ◮ Then H has finite index if and only if every word of L can be

read in ΓL(H) starting from the base vertex

◮ This is decidable ◮ In that case, ΓL(H) is a subgraph of the (finite) Schreier

graph, with all the vertices

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic

Thank you for your attention!

Pascal Weil Stallings graphs for quasi-convex subgps of hyperbolic