On the isomorphism problem for relatively hyperbolic groups

Transcription

1 On the isomorphism problem for relatively hyperbolic groups Nicholas Touikan (joint work with François Dahmani) CMS Winter 2012 Geometrical Group Theory session December

2 The isomorphism problem Given two finite presentations x 1,..., x n r 1,... r m ; y 1,..., y p s 1,..., s q is it possible to decide if they represent isomorphic groups?

3 The isomorphism problem Given two finite presentations x 1,..., x n r 1,... r m ; y 1,..., y p s 1,..., s q is it possible to decide if they represent isomorphic groups? In general no (Adian, Rabin late 1950s.)

4 The isomorphism problem Given two finite presentations x 1,..., x n r 1,... r m ; y 1,..., y p s 1,..., s q is it possible to decide if they represent isomorphic groups? In general no (Adian, Rabin late 1950s.) That being said this is still a reasonnable question to ask if the presentation are known to lie a restricted class of groups. For example if both presentations are known to be of abelian groups, then we can straightforwardly decide if these two presentations give isomorphic groups.

5 Previous results Grunewald and Segal solve the isomorphism problem for f.g. nilpotent groups (1980).

6 Previous results Grunewald and Segal solve the isomorphism problem for f.g. nilpotent groups (1980). Segal solves the isomorphism problem for polycyclic-by-finite groups (1990).

7 Previous results Grunewald and Segal solve the isomorphism problem for f.g. nilpotent groups (1980). Segal solves the isomorphism problem for polycyclic-by-finite groups (1990). Sela solves the isomorphism problem for rigid torsion-free hyperbolic groups (1995).

8 Previous results Grunewald and Segal solve the isomorphism problem for f.g. nilpotent groups (1980). Segal solves the isomorphism problem for polycyclic-by-finite groups (1990). Sela solves the isomorphism problem for rigid torsion-free hyperbolic groups (1995). Bumagin, Kharlampovich, and Miasnikov solve the isomorphism problem for f.g. fully residually free, or limit, groups (2007).

9 Previous results Grunewald and Segal solve the isomorphism problem for f.g. nilpotent groups (1980). Segal solves the isomorphism problem for polycyclic-by-finite groups (1990). Sela solves the isomorphism problem for rigid torsion-free hyperbolic groups (1995). Bumagin, Kharlampovich, and Miasnikov solve the isomorphism problem for f.g. fully residually free, or limit, groups (2007). Dahamni and Groves solve the isomorphism problem for toral relatively hyperbolic groups (2008), this class includes torsion-free hyperbolic groups and limit groups.

10 Previous results Grunewald and Segal solve the isomorphism problem for f.g. nilpotent groups (1980). Segal solves the isomorphism problem for polycyclic-by-finite groups (1990). Sela solves the isomorphism problem for rigid torsion-free hyperbolic groups (1995). Bumagin, Kharlampovich, and Miasnikov solve the isomorphism problem for f.g. fully residually free, or limit, groups (2007). Dahamni and Groves solve the isomorphism problem for toral relatively hyperbolic groups (2008), this class includes torsion-free hyperbolic groups and limit groups. Dahmani and Guirardel solve the isomorphism problem for all hyperbolic groups; even those with torsion (2011).

11 A new result Theorem (Dahmani-T) There is an algorithm which given two presentations x 1,..., x n r 1,... r m ; y 1,..., y p s 1,..., s q of torsion-free groups that are known to be hyperbolic relative to nilpotent groups, will decide if they are isomorphic.

12 Some terminology: Peripheral structures and relative hyperbolicity Let G be a group, a peripheral structure P on G is a (possibly empty) union of finitely many conjugacy classes: where [P i ] = {g 1 P i g g G}. P = [P 1 ] [P s ]

13 Some terminology: Peripheral structures and relative hyperbolicity Let G be a group, a peripheral structure P on G is a (possibly empty) union of finitely many conjugacy classes: P = [P 1 ] [P s ] where [P i ] = {g 1 P i g g G}. Each subgroup H of some P P is called parabolic, in particular P is the set of maximal parabolic subgroups.

14 Some terminology: Peripheral structures and relative hyperbolicity Let G be a group, a peripheral structure P on G is a (possibly empty) union of finitely many conjugacy classes: P = [P 1 ] [P s ] where [P i ] = {g 1 P i g g G}. Each subgroup H of some P P is called parabolic, in particular P is the set of maximal parabolic subgroups. A relatively hyperbolic group is a pair (G, P) which, informally, looks like a hyperbolic group outside the parabolic subgroups.

15 Some terminology: Peripheral structures and relative hyperbolicity Let G be a group, a peripheral structure P on G is a (possibly empty) union of finitely many conjugacy classes: P = [P 1 ] [P s ] where [P i ] = {g 1 P i g g G}. Each subgroup H of some P P is called parabolic, in particular P is the set of maximal parabolic subgroups. A relatively hyperbolic group is a pair (G, P) which, informally, looks like a hyperbolic group outside the parabolic subgroups. For example: (Z 2 Z 2, P) where P is the set of maximal non-cyclic abelian subgroups is a relatively hyperbolic group.

16 Some terminology: Peripheral structures and relative hyperbolicity Let G be a group, a peripheral structure P on G is a (possibly empty) union of finitely many conjugacy classes: P = [P 1 ] [P s ] where [P i ] = {g 1 P i g g G}. Each subgroup H of some P P is called parabolic, in particular P is the set of maximal parabolic subgroups. A relatively hyperbolic group is a pair (G, P) which, informally, looks like a hyperbolic group outside the parabolic subgroups. For example: (Z 2 Z 2, P) where P is the set of maximal non-cyclic abelian subgroups is a relatively hyperbolic group. We also say that G is hyperbolic relative to P.

17 Some terminology: Elementary splittings An elementary splitting of (G, P) is an (essential) decomposition of G as the fundamental group of a graph of groups such that: Edge groups are parabolic, virtually cyclic, or finite, Each P P is conjugate into a vertex groups

18 Some terminology: Elementary splittings An elementary splitting of (G, P) is an (essential) decomposition of G as the fundamental group of a graph of groups such that: Edge groups are parabolic, virtually cyclic, or finite, Each P P is conjugate into a vertex groups For example: G = A C B; G = A B D where C is parabolic and each P P is conjugate into A, B, D.

19 Some terminology: Elementary splittings An elementary splitting of (G, P) is an (essential) decomposition of G as the fundamental group of a graph of groups such that: Edge groups are parabolic, virtually cyclic, or finite, Each P P is conjugate into a vertex groups For example: G = A C B; G = A B D where C is parabolic and each P P is conjugate into A, B, D. We say that a relatively hyperbolic group (G, P) is rigid if it is not virtually cyclic and it admits no elementary splittings.

20 The general strategy for the isomorphism problem Let (G, P) and (H, Q) (given by presentations) be relatively hyperbolic (possibly with P, Q = ).

21 The general strategy for the isomorphism problem Let (G, P) and (H, Q) (given by presentations) be relatively hyperbolic (possibly with P, Q = ). Step 1: If (G, P) and (H, Q) are one-ended (relative to P, Q) construct canonical elementary JSJ splittings for (G, P) and (H, Q).

22 The general strategy for the isomorphism problem Let (G, P) and (H, Q) (given by presentations) be relatively hyperbolic (possibly with P, Q = ). Step 1: If (G, P) and (H, Q) are one-ended (relative to P, Q) construct canonical elementary JSJ splittings for (G, P) and (H, Q).

23 The general strategy for the isomorphism problem Let (G, P) and (H, Q) (given by presentations) be relatively hyperbolic (possibly with P, Q = ). Step 1: If (G, P) and (H, Q) are one-ended (relative to P, Q) construct canonical elementary JSJ splittings for (G, P) and (H, Q). So now we check if the graphs of groups look the same. I.e. if the underlying graphs are isomorphic.

24 The general strategy for the isomorphism problem Step 2: The black vertex groups are either periperal or virtually cyclic. We enlarge the peripheral structures P ˆP, Q ˆQ so that the black vertex groups are peripheral. The white vertex groups are either QH (or surface type) or (w.r.t. the natural induced rel. hyp. structure) rigid. We now solve the isomorphism problem for the vertex groups.

25 The general strategy for the isomorphism problem Step 2: The black vertex groups are either periperal or virtually cyclic. We enlarge the peripheral structures P ˆP, Q ˆQ so that the black vertex groups are peripheral. The white vertex groups are either QH (or surface type) or (w.r.t. the natural induced rel. hyp. structure) rigid. We now solve the isomorphism problem for the vertex groups. Step 3: If the previous steps went though, see if the graphs of groups assemble to give isomorphic groups.

26 The isomorphism problem: the rigid case We give a sketch of Sela s algorithm to solve the isomorphism problem for rigid hyperbolic groups (i.e. P = ).

27 The isomorphism problem: the rigid case We give a sketch of Sela s algorithm to solve the isomorphism problem for rigid hyperbolic groups (i.e. P = ).Let Γ x 1,..., x n r 1,... r m ; y 1,..., y p s 1,..., s q be rigid hyperbolic groups. The theory of actions on R-trees tells us that either: There are monomorphisms Γ and Γ, which by co-hopficity, implies that Γ; or

28 The isomorphism problem: the rigid case We give a sketch of Sela s algorithm to solve the isomorphism problem for rigid hyperbolic groups (i.e. P = ).Let Γ x 1,..., x n r 1,... r m ; y 1,..., y p s 1,..., s q be rigid hyperbolic groups. The theory of actions on R-trees tells us that either: There are monomorphisms Γ and Γ, which by co-hopficity, implies that Γ; or W.l.o.g. there is some finite set F Γ such that for every f Hom(Γ, ) f F is not injective. We call such a set an obstruction from Γ to.

29 The isomorphism problem: the rigid case We now have two processes. Process 1: Enumerate via Tietze transformations all presentations isomorphic to x 1,..., x n r 1,... r m. If our presentation y 1,..., y p s 1,..., s q of appear stop and output Γ.

30 The isomorphism problem: the rigid case We now have two processes. Process 1: Enumerate via Tietze transformations all presentations isomorphic to x 1,..., x n r 1,... r m. If our presentation y 1,..., y p s 1,..., s q of appear stop and output Γ. Process 2: Look for obstructions from to Γ and vice versa. Take F 1 F 2..., E 1 E 2... finite exhaustions (group elements represented as words) of Γ \ {1}, \ {1} and check the truth of the first order sentences

31 The isomorphism problem: the rigid case We now have two processes. Process 1: Enumerate via Tietze transformations all presentations isomorphic to x 1,..., x n r 1,... r m. If our presentation y 1,..., y p s 1,..., s q of appear stop and output Γ. Process 2: Look for obstructions from to Γ and vice versa. Take F 1 F 2..., E 1 E 2... finite exhaustions (group elements represented as words) of Γ \ {1}, \ {1} and check the truth of the first order sentences Γ = y 1,..., y p ( q i=1 = x 1,..., x n ( m i=1 ) ( ) s i (y 1,..., y p ) = 1 f (y 1,..., y p ) 1 f E k ) ( ) r i (x 1,..., x n ) = 1 e(x 1,..., x n ) 1 e F k

32 The isomorphism problem: the rigid case Process 2 ends if one of these sentences is false, in which case we have found an obstruction and conclude Γ.

33 The isomorphism problem: the rigid case Process 2 ends if one of these sentences is false, in which case we have found an obstruction and conclude Γ. Process 2 is doable because by results Makanin, Rips-Sela, and Sela the existential theory of torsion free hyperbolic groups is decidable.

34 The isomorphism problem: the rigid case Process 2 ends if one of these sentences is false, in which case we have found an obstruction and conclude Γ. Process 2 is doable because by results Makanin, Rips-Sela, and Sela the existential theory of torsion free hyperbolic groups is decidable. Dahmani showed that the universal theory of a torsion free relatively hyperbolic groups is decidable provided the existential theory is decidable in the parabolic subgroups.

35 The isomorphism problem: the rigid case Process 2 ends if one of these sentences is false, in which case we have found an obstruction and conclude Γ. Process 2 is doable because by results Makanin, Rips-Sela, and Sela the existential theory of torsion free hyperbolic groups is decidable. Dahmani showed that the universal theory of a torsion free relatively hyperbolic groups is decidable provided the existential theory is decidable in the parabolic subgroups. Dahmani and Guirardel then showed that this is decidable for virtually free groups and used this for the case of arbitrary hyperbolic groups.

36 The isomorphism problem: an existential crisis Theorem (Roman kov 1979) The existential theory of nilpotent groups is undecidable.

37 The isomorphism problem: an existential crisis Theorem (Roman kov 1979) The existential theory of nilpotent groups is undecidable. Theorem (Truss 1995) Let G a free nilpotent group of rank 2 and class 3. Then the problem of determining the solubility of general systems of euqations in G is undecidable.

38 The isomorphism problem: an existential crisis Theorem (Roman kov 1979) The existential theory of nilpotent groups is undecidable. Theorem (Truss 1995) Let G a free nilpotent group of rank 2 and class 3. Then the problem of determining the solubility of general systems of euqations in G is undecidable. To make things worse Step 1 in the previous methods (compute the JSJ) also heavily makes use of equationnal methods.

39 Dehn fillings. Let (G, P) and (H, Q) relatively hyperbolic groups with P = [P 1 ] [P s ], Q = [Q 1 ] [Q s ]. With Q, P collections of residually finite groups.

40 Dehn fillings. Let (G, P) and (H, Q) relatively hyperbolic groups with P = [P 1 ] [P s ], Q = [Q 1 ] [Q s ]. With Q, P collections of residually finite groups. Let K n = P (n) 1,..., P(n) s G where P (n) i = H and define L n H similarly. [P i :H] n

41 Dehn fillings. Let (G, P) and (H, Q) relatively hyperbolic groups with P = [P 1 ] [P s ], Q = [Q 1 ] [Q s ]. With Q, P collections of residually finite groups. Let K n = P (n) 1,..., P(n) s G where P (n) i = H [P i :H] n and define L n H similarly. We call G/K n the n th characteristic Dehn filling. For n >> 0 G/K n is hyperbolic relative to P n = s [P i /(P i K n )] and P i /K n P i /P (n) i=1 This is due to Osin and Groves-Manning. i.

42 Dahmani and Guirardel s approach to rigid case. Dahmani and Guirardel have announced the following: Let (G, P) (H, Q) be f.p. rigid relatively hyperbolic groups with residually finite peripheral subgroups.

43 Dahmani and Guirardel s approach to rigid case. Dahmani and Guirardel have announced the following: Let (G, P) (H, Q) be f.p. rigid relatively hyperbolic groups with residually finite peripheral subgroups. Then if there exist α 1, α 2,... such that for all 0 << n < m (G, P) (H, Q) (G/K m, P m ) α m (H/L m, Q m ) (G/K n, P n ) α n (H/L n, Q n )

44 Dahmani and Guirardel s approach to rigid case. Dahmani and Guirardel have announced the following: Let (G, P) (H, Q) be f.p. rigid relatively hyperbolic groups with residually finite peripheral subgroups. Then if there exist α 1, α 2,... such that for all 0 << n < m (G, P) α (H, Q) (G/K m, P m ) α m (H/L m, Q m ) (G/K n, P n ) α n (H/L n, Q n )

45 Dahmani and Guirardel s approach to rigid case. This essentially enables Dahmani and Guirardel to reduce the isomorphism problem for rigid relatively hyperbolic groups (with r.f. parabolics, and some other technical criteria) to the isomorphism problem for hyperbolic groups with torsion.

46 Finding the JSJ

47 Finding the JSJ With Nicholas track finding algorithm (arxiv 2011) it is possible to decide whether a torsion free relatively hyperbolic group (G, P) admits an essential elementary splitting without having to resort to solving equations, provided P belongs to a class of algorithmically tractable groups.

48 Finding the JSJ Following Guirardel and Levitt we chose our canonical JSJ decomposition to be dual to a (G, P)-tree T c which is obtained as the tree of cyclinders for co-elementarity of some tree (G, P)-tree T living in the (G, P) JSJ deformation space.

49 Finding the JSJ Following Guirardel and Levitt we chose our canonical JSJ decomposition to be dual to a (G, P)-tree T c which is obtained as the tree of cyclinders for co-elementarity of some tree (G, P)-tree T living in the (G, P) JSJ deformation space. Their (arxiv 2010) result that T c is also maximal universally compatible, combined with Nicholas result gives a relatively straightforward algorithm to compute the canonical JSJ decomposition.

50 Finding the JSJ Following Guirardel and Levitt we chose our canonical JSJ decomposition to be dual to a (G, P)-tree T c which is obtained as the tree of cyclinders for co-elementarity of some tree (G, P)-tree T living in the (G, P) JSJ deformation space. Their (arxiv 2010) result that T c is also maximal universally compatible, combined with Nicholas result gives a relatively straightforward algorithm to compute the canonical JSJ decomposition. Theorem (Dahmani-T) If (G, P) is relatively hyperbolic with P algorithmically tractable and effectively coherent, then we can compute the canonical JSJ decomposition of (G, P).

51 Putting things together (general nonsense) Let X, X graphs of groups with underlying bipartite directed graphs X, X. Let x x be an isomorphism from X to X. Suppose for each v V (X ) we have some ψ v : Γ v, Γ v.

52 Putting things together (general nonsense) Let X, X graphs of groups with underlying bipartite directed graphs X, X. Let x x be an isomorphism from X to X. Suppose for each v V (X ) we have some ψ v : Γ v, Γ v. u e v u e v

53 Putting things together (general nonsense) Let X, X graphs of groups with underlying bipartite directed graphs X, X. Let x x be an isomorphism from X to X. Suppose for each v V (X ) we have some ψ v : Γ v, Γ v. Γ u t e u Γ e e i e v Γ v u e v Γ u t e Γ e i e Γ v

54 Putting things together (general nonsense) Let X, X graphs of groups with underlying bipartite directed graphs X, X. Let x x be an isomorphism from X to X. Suppose for each v V (X ) we have some ψ v : Γ v, Γ v. Γ u t e u Γ e e i e v Γ v ψ u u e v ψ v Γ u t e Γ e i e Γ v

55 Putting things together (general nonsense) Then the map X X given by x x is induced by an isomorphism π 1 (X) π 1 (X ) if and only if there exist for each vertex group Γ w an extension adjustement α w Aut(Γ w ) and for each edge e elements g e Γ 0(e), g e + Γ t(e). Such that: Γ u t e Γ e i e Γ v ψ u Γ u Γ v ψ v Γ u t e Γ e i e Γ v

56 Putting things together (general nonsense) Then the map X X given by x x is induced by an isomorphism π 1 (X) π 1 (X ) if and only if there exist for each vertex group Γ w an extension adjustement α w Aut(Γ w ) and for each edge e elements g e Γ 0(e), g e + Γ t(e). Such that: Γ u t e Γ e i e Γ v ψ u Γ u ψ v Γ v α u Γ u α v Γ v Γ u t e Γ e i e Γ v

57 Putting things together (general nonsense) Then the map X X given by x x is induced by an isomorphism π 1 (X) π 1 (X ) if and only if there exist for each vertex group Γ w an extension adjustement α w Aut(Γ w ) and for each edge e elements g e Γ 0(e), g e + Γ t(e). Such that: Γ u t e Γ e i e Γ v ψ u Γ u ψ v Γ v α u Γ u α v Γ v ad ge Γ u t e Γ e i e Γ v ad ge +

58 Putting things together (general nonsense) Then the map X X given by x x is induced by an isomorphism π 1 (X) π 1 (X ) if and only if there exist for each vertex group Γ w an extension adjustement α w Aut(Γ w ) and for each edge e elements g e Γ 0(e), g e + Γ t(e). Such that: Γ u t e s Γ e i e Γ v ψ u Γ u ψ v Γ v α u Γ u α v Γ v ad ge Γ u t e Γ e i e Γ v ad ge +

59 Putting things together (general nonsense) Then the map X X given by x x is induced by an isomorphism π 1 (X) π 1 (X ) if and only if there exist for each vertex group Γ w an extension adjustement α w Aut(Γ w ) and for each edge e elements g e Γ 0(e), g e + Γ t(e). Such that: Γ u t e s Γ e i e Γ v ψ u Γ u ψ v Γ v α u Γ u α v Γ v ad ge Γ u t e i e Γ e s Γ v ad ge +

60 Putting things together (general nonsense) Then the map X X given by x x is induced by an isomorphism π 1 (X) π 1 (X ) if and only if there exist for each vertex group Γ w an extension adjustement α w Aut(Γ w ) and for each edge e elements g e Γ 0(e), g e + Γ t(e). Such that: Γ u t e s Γ e i e Γ v ψ u Γ u ψ v Γ v α u Γ u α v Γ v ad ge Γ u t e Γ e s s i e Γ v ad ge +

61 Putting things together (general nonsense) Then the map X X given by x x is induced by an isomorphism π 1 (X) π 1 (X ) if and only if there exist for each vertex group Γ w an extension adjustement α w Aut(Γ w ) and for each edge e elements g e Γ 0(e), g e + Γ t(e). Such that: Γ u t e s Γ e i e Γ v ψ u Γ u ψ v Γ v α u Γ u α v Γ v ad ge Γ u t e Γ e s = s i e Γ v ad ge +

62 Putting things together: outer automorphisms Let (G, P) relatively hyperbolic with P = [P 1 ] [P s ], clasically we have Out(G) = Aut(G)/Inn(G)

63 Putting things together: outer automorphisms Let (G, P) relatively hyperbolic with P = [P 1 ] [P s ], clasically we have Out(G) = Aut(G)/Inn(G) we further define Out(G, P) = {[α] Out(G) g 1,..., g s G, α(p i ) = g 1 i P i g}.

64 Putting things together: outer automorphisms Let (G, P) relatively hyperbolic with P = [P 1 ] [P s ], clasically we have Out(G) = Aut(G)/Inn(G) we further define Out(G, P) = {[α] Out(G) g 1,..., g s G, α(p i ) = g 1 i P i g}. Fact 1: For each P i (since these are self-normalized) there is a well defined restriction Out(G, P) Out(P i )

65 Putting things together: outer automorphisms Let (G, P) relatively hyperbolic with P = [P 1 ] [P s ], clasically we have Out(G) = Aut(G)/Inn(G) we further define Out(G, P) = {[α] Out(G) g 1,..., g s G, α(p i ) = g 1 i P i g}. Fact 1: For each P i (since these are self-normalized) there is a well defined restriction Out(G, P) Out(P i ) Fact 2: If (G, P) is rigid then Out(G, P) is finite..

66 Putting things together: the relatively hyperbolic case Each white vertex groups Γ v comes equipped with an induced rel. hyp. structure (Γ v, P v ) in which the images of the edge groups are parabolic subgroups. and such that (Γ v, P v ) is rigid.

67 Putting things together: the relatively hyperbolic case Each white vertex groups Γ v comes equipped with an induced rel. hyp. structure (Γ v, P v ) in which the images of the edge groups are parabolic subgroups. and such that (Γ v, P v ) is rigid. So for any tuple of generators S = (s 1,..., s n ) its images = g 1 e ψ + v α v (S)g e + S α v,g e +

68 Putting things together: the relatively hyperbolic case Each white vertex groups Γ v comes equipped with an induced rel. hyp. structure (Γ v, P v ) in which the images of the edge groups are parabolic subgroups. and such that (Γ v, P v ) is rigid. So for any tuple of generators S = (s 1,..., s n ) its images = g 1 e ψ + v α v (S)g e + S α v,g e + Γ u t e Γ e i e Γ v ψ u Γ u Γ v ψ v Γ v α v Γ u t e Γ e i e Γ v ad ge +.

69 Putting things together: the relatively hyperbolic case Each white vertex groups Γ v comes equipped with an induced rel. hyp. structure (Γ v, P v ) in which the images of the edge groups are parabolic subgroups. and such that (Γ v, P v ) is rigid. So for any tuple of generators S = (s 1,..., s n ) its images = g 1 e ψ + v α v (S)g e + S α v,g e + Γ u t e S Γ e i e Γ v ψ u Γ u Γ v ψ v Γ v α v Γ u t e Γ e i e Γ v ad ge + S α v,g e + Take on only finitely many possible values up to conjugacy in Γ e.

70 Putting things together: the relatively hyperbolic case So let T be defined as:

71 Putting things together: the relatively hyperbolic case So let T be defined as: Γ u t e S Γ e i e Γ v T ψ u Γ u ψ v Γ v α v Γ v Γ u t e Γ e i e Γ v ad ge +

72 Putting things together: the relatively hyperbolic case So let T be defined as: Γ u t e S Γ e i e Γ v T ψ u Γ u ψ v Γ v α v Γ v Γ u S t e Γ e i e Γ v ad ge +

73 Putting things together: the relatively hyperbolic case So let T be defined as: Γ u t e S Γ e i e Γ v T ψ u Γ u α T,S Γ u ψ v Γ v α v Γ v ad gt,s Γ u S t e Γ e i e Γ v ad ge + We need to find α T,S, g T,S so that ad GT,S α T,S (T ) = S.

74 Putting things together: the relatively hyperbolic case So let T be defined as: Γ u t e S Γ e i e Γ v T ψ u Γ u α T,S Γ u ψ v Γ v α v Γ v ad gt,s Γ u S t e Γ e i e Γ v ad ge + We need to find α T,S, g T,S so that ad GT,S α T,S (T ) = S. Things get more complicated because black vertex groups can have multiple incident edge groups..

75 Putting things together: the mixed Whitehead problem Let (S 1,..., S n ), (T 1,..., T n ) be a tuple of tuples of elements in G the mixed Whitehead problem (MWHP) asks whether there exists some α Aut(G) and g 1,..., g n G such that g 1 i α(s i )g i = T i.

76 Putting things together: the mixed Whitehead problem Let (S 1,..., S n ), (T 1,..., T n ) be a tuple of tuples of elements in G the mixed Whitehead problem (MWHP) asks whether there exists some α Aut(G) and g 1,..., g n G such that g 1 i α(s i )g i = T i. Theorem (Dahmani-T) The MWHP is solvable in the class of f.g. nilpotent groups.

77 Putting things together: the mixed Whitehead problem Let (S 1,..., S n ), (T 1,..., T n ) be a tuple of tuples of elements in G the mixed Whitehead problem (MWHP) asks whether there exists some α Aut(G) and g 1,..., g n G such that g 1 i α(s i )g i = T i. Theorem (Dahmani-T) The MWHP is solvable in the class of f.g. nilpotent groups. The proof heavily relies on the algorithmic methods developped by Grunewald and Segal to solve orbit problems for rational actions of arithmetic groups.

78 Putting things together: the mixed Whitehead problem Let (S 1,..., S n ), (T 1,..., T n ) be a tuple of tuples of elements in G the mixed Whitehead problem (MWHP) asks whether there exists some α Aut(G) and g 1,..., g n G such that g 1 i α(s i )g i = T i. Theorem (Dahmani-T) The MWHP is solvable in the class of f.g. nilpotent groups. The proof heavily relies on the algorithmic methods developped by Grunewald and Segal to solve orbit problems for rational actions of arithmetic groups. Bogopolski and Ventura also proved the MWHP (and coined the term) for t.f. hyperbolic groups.

79 Putting things together: reduction to the mixed Whitehead problem Suppose it was possible for every white vertex v group to construct the finite list of images S α v,g = g 1 e + e α + v ψ v (S)g e + up to conjugacy in Γ u. Γ u t e S Γ e i e Γ v ψ u Γ u Γ v ψ v Γ v α v Γ u S α v,g e + t e Γ e i e Γ v ad ge +

80 Putting things together: reduction to the mixed Whitehead problem Suppose it was possible for every white vertex v group to construct the finite list of images S α v,g = g 1 e + e α + v ψ v (S)g e + up to conjugacy in Γ u. Γ u t e S Γ e i e Γ v ψ u Γ u Γ v ψ v Γ v α v Γ u t e Γ e i e Γ v ad ge + S α v,g e + Then the isomorphism problem can be reduced to finitely many instances of the MWHP in the black vertex groups.

81 Putting things together: orbit computations Let (G, P) with P = [P 1 ] [P s ] be relatively hyperbolic, the we have the well defined restriction map Out(G, P) Out(P i ).

82 Putting things together: orbit computations Let (G, P) with P = [P 1 ] [P s ] be relatively hyperbolic, the we have the well defined restriction map Out(G, P) Out(P i ). If (G, P) is rigid then the image T = r(out(g, P)) of r : Out(G, P) n Out(P i ) i=1 is finite.

83 Putting things together: orbit computations Let (G, P) with P = [P 1 ] [P s ] be relatively hyperbolic, the we have the well defined restriction map Out(G, P) Out(P i ). If (G, P) is rigid then the image T = r(out(g, P)) of r : Out(G, P) n Out(P i ) i=1 is finite. For our purposes it is in fact sufficient to compute T, i.e. find a set L = {α,..., α r } Aut(G, P) which give representatives of T, to construct the finite list of images needed for our reduction to the MWHP.

84 Putting things together: orbit computations Let (G, P) with P = [P 1 ] [P s ] be relatively hyperbolic, the we have the well defined restriction map Out(G, P) Out(P i ). If (G, P) is rigid then the image T = r(out(g, P)) of r : Out(G, P) n Out(P i ) i=1 is finite. For our purposes it is in fact sufficient to compute T, i.e. find a set L = {α,..., α r } Aut(G, P) which give representatives of T, to construct the finite list of images needed for our reduction to the MWHP. By the way we do not know how to compute Out(G, P) (all known methods involve equationnal methods.)

85 A digression:separating torsion in Out(P) Let P be a group, we say that congruences of P separate the torsion in Out(P) if there is some finite index characteristic subgroup P 0 P such that in the natural map π P0 : Out(P) Out(P/P o ) every finite order [α] Out(P) survives, equivalently the kernel of π P0 is torsion free.

86 A digression:separating torsion in Out(P) Let P be a group, we say that congruences of P separate the torsion in Out(P) if there is some finite index characteristic subgroup P 0 P such that in the natural map π P0 : Out(P) Out(P/P o ) every finite order [α] Out(P) survives, equivalently the kernel of π P0 is torsion free. A celebrated example is for P = Z m, then every finite order element of GL(m, Z) survives in GL(m, Z/nZ) for n sufficiently large.

87 A digression:separating torsion in Out(P) We say that congruences of P effectively separate the torsion in Out(P) if there is an algorithm which finds a finite index characteristic subgroup P 0 which is deep enough so that ker(π P0 ) is torsion free.

88 A digression:separating torsion in Out(P) We say that congruences of P effectively separate the torsion in Out(P) if there is an algorithm which finds a finite index characteristic subgroup P 0 which is deep enough so that ker(π P0 ) is torsion free. Theorem (Segal, private communication) If P polycyclic-by-finite then congruences of P separate the torsion in Out(P). Theorem (Dahmani-T) There is a uniform algorithm for all f.g. nilpotent groups so that if N is nilpotent then congruences of N effectively separate the torsion in Out(N).

89 Putting things together: orbit computations So going back to our (G, P) with P = [P 1 ] [P s ] if congruences in P i effectively separate the torsion in Out(P i ) then we can pick N so large that for n N,

90 Putting things together: orbit computations So going back to our (G, P) with P = [P 1 ] [P s ] if congruences in P i effectively separate the torsion in Out(P i ) then we can pick N so large that for n N, The n th characteristic Dehn filling (G/K n, P n ) is relatively hyperbolic with P n = [P 1 /K n ] [P s /K n ].

91 Putting things together: orbit computations So going back to our (G, P) with P = [P 1 ] [P s ] if congruences in P i effectively separate the torsion in Out(P i ) then we can pick N so large that for n N, The n th characteristic Dehn filling (G/K n, P n ) is relatively hyperbolic with P n = [P 1 /K n ] [P s /K n ]. The map induced by T n Out(P i /K n ) i=1 Out(G, P) Out(P i ) Out(G/K n, P n ) Out(P i /K n ) is injective.

92 Computing T The computability of T now follows from an enumeration argument that uses the Dahmani-Guirardel isomorphism lifting principle,

93 Computing T The computability of T now follows from an enumeration argument that uses the Dahmani-Guirardel isomorphism lifting principle,i.e. (G, P) α (H, Q) (G/K m, P m ) α m (H/L m, Q m ) (G/K n, P n ) α n (H/L n, Q n ) and the fact that automorphisms of a group with solvable word problem are enumerable.

94 The criteria A class C of groups is said to be algorithmically tractable if The finite presentation of the groups in C are recursively enumerable. Limit groups, certain classes of small cancellation groups (i.e. those with some type of effective coherence), and polycyclic-by-finite groups lie in this class.

95 The criteria A class C of groups is said to be algorithmically tractable if The finite presentation of the groups in C are recursively enumerable. The conjugacy problem is uniformly solvable in C. Limit groups, certain classes of small cancellation groups (i.e. those with some type of effective coherence), and polycyclic-by-finite groups lie in this class.

96 The criteria A class C of groups is said to be algorithmically tractable if The finite presentation of the groups in C are recursively enumerable. The conjugacy problem is uniformly solvable in C. The generation problem is uniformly solvable in C. Limit groups, certain classes of small cancellation groups (i.e. those with some type of effective coherence), and polycyclic-by-finite groups lie in this class.

97 Theorem (Dahmani-T) There is an algorithm which takes explicit presentations of torsion free relatively hyperbolic groups (G, [P 1 ] [P s ]), (H, [Q 1 ] [Q s ]) and provided the P i, Q j lie in a class C of groups 1. that is algorithmically tractable, 2. that is uniformly effectively coherent, 3. in which we can solve the isomorphism problem, 4. in which the mixed Whitehead problem is uniformly solvable, and 5. in which congruence effectively separate torsion; then the algorithm decides whether or not the two groups are isomorphic. (G, [P 1 ] [P s ]), (H, [Q 1 ] [Q s ])