Makanin-Razborov diagrams for relatively hyperbolic groups

Transcription

1 Makanin-Razborov diagrams for relatively hyperbolic groups Nicholas Touikan (joint with Inna Bumagin) AMS Fall Eastern Sectional Meeting, Bowdoin 2016 Special GGT Session

2 G source group. Γ target group. What is Hom (G, Γ)? Example F n free group. Hom (F n, Γ) = Γ n

3 G = x 1,..., x n r 1 (x 1,..., x n ), r 2 (x 1,..., x n ),... f Hom (G, Γ) with f (x i ) = γ i Γ is a solution to the (possibly infinite) system of equations. r 1 (x 1,..., x n ) = Γ 1 r 2 (x 1,..., x n ) = Γ 1... Being able to uniformly describe such sets is a feature of the target group Γ.

4 G = π 1 ( ) = x, y, z, t, u, v [x, y][z, t][u, v] = 1 Γ = π 1 ( ) = a, b, c, d [a, b][c, d] = 1

5 G Γ

6 G Γ π 1 G Γ id π 15 id G Γ

7 G Γ π 1 G Γ id mcg(σ 3 ) π 15 id G Γ

8 G Γ π 1 G Γ id mcg(σ 3 ) π 15 id Inn(Γ) G Γ

9 G Γ π 1 G Γ id mcg(σ 3 ) π 15 id Inn(Γ) G π p Γ Z 2 Γ

10 G Γ π 1 G Γ id mcg(σ 3 ) π 15 id Inn(Γ) G π p Γ Z 2 Γ id Γ Z 2 t Γ t Γ

11 G Γ π 1 G Γ id mcg(σ 3 ) π 15 id Inn(Γ) G π p Γ Z 2 Γ id Γ Z 2 t Γ t Γ id Γ t Γ

12 G Γ π 1 G Γ id mcg(σ 3 ) π 15 id Inn(Γ) G π p Γ Z 2 Γ id Γ Z 2 t Γ t Γ id Γ t Γ F 3

13 G Γ π 1 G Γ id mcg(σ 3 ) π 15 id Inn(Γ) G π p Γ Z 2 Γ id Γ Z 2 t Γ t Γ id Γ t Γ π ab t SL(6, Z) Z 6 F 3 (a, b, c, d, e, f ) t a t

14 Makanin-Razborov diagrams have the following features: (i) They contain finitely many groups, and all homomorphisms in the diagram, except automorphisms and free variable specializations at the last level are fixed. (ii) All of Hom (G, Γ) is obtained by composition of these homomorphisms and certain types of automorphisms of the groups.

15 The Groves-Manning definition.

16 The Groves-Manning definition.

17 The Groves-Manning definition.

18 The Groves-Manning definition.

19 The Groves-Manning definition.

20 The Groves-Manning definition.

21 The Groves-Manning definition.

22 Theorem Let Γ = (Γ; P 1,..., P n ) be a relatively hyperbolic group such that each parabolic subgroup P i is slender, i.e. every subgroup is finitely generated, and equationally Noetherian. Then for every finitely generated G there exists a Makanin-Razborov diagram that encodes Hom (G, Γ). The terminal groups decompose as graphs of groups with parabolic or bounded subgroups. The images of the bounded subgroups are fixed up to conjugacy in Γ. The parabolic groups are finitely generated.

23 Theorem Let Γ = (Γ; P 1,..., P n ) be a relatively hyperbolic group such that each parabolic subgroup P i is slender, i.e. every subgroup is finitely generated, and equationally Noetherian. Then for every finitely generated G there exists a Makanin-Razborov diagram that encodes Hom (G, Γ). The terminal groups decompose as graphs of groups with parabolic or bounded subgroups. The images of the bounded subgroups are fixed up to conjugacy in Γ. The parabolic groups are finitely generated. The parabolic vertex groups play the role of free variables.

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26 This reduces the description of Hom (G, Γ) to the description of finitely many sets Hom (G k, P i ). Such reduction results for relatively hyperbolic groups abound: 1. If the word problem is decidable in P i then it is also in Γ (Farb.) 2. If the conjugacy problem is decidable in P i then it is also in Γ (Bumagin.)

27 Makanin decides if a system of equations over a free group has a (non-trivial) solution. Razborov gives a description of all such solutions. Kharlampovich-Miasnikov and, independently, Sela give diagrams where the groups inside are fully residually free or limit groups. Sela makes diagrams for torsion-free hyperbolic groups. Also proves that hyperbolic groups are equationally Noetherian. Alibegovic makes diagrams for limit groups (special case of toral relatively hyperbolic groups.) Introduces bending move (inspired by Thurston) and works with cusped space. Groves makes diagrams for toral relatively hyperbolic groups.

28 Groves uses the global geometry of toral relatively hyperbolic groups. In particular asymptotic cone contains R n s. Alibegovic cusps off the Z n subgroups, but the bending move only works if the parabolics are abelian. And then there s torsion...

29 Reinfeldt and Weidmann construct diagrams for all hyperbolic groups. I.e. they deal with finite-by-(2-orbifold) groups and virtually abelian groups. Jaligot and Sela construct diagrams for free products. They introduce elliptic structure. Drutu and Sapir, studying actions on tree graded spaces, develop tools to study Hom (G, Γ), but some of these actions are too wild to use the shortening argument.

30 Reinfeldt and Weidmann construct diagrams for all hyperbolic groups. I.e. they deal with finite-by-(2-orbifold) groups and virtually abelian groups. Jaligot and Sela construct diagrams for free products. They introduce elliptic structure. Drutu and Sapir, studying actions on tree graded spaces, develop tools to study Hom (G, Γ), but some of these actions are too wild to use the shortening argument. Casals-Ruiz and Kazachkov conctruct diagrams when Γ is partially commutative (or a RAAG.) Jez, using a new technique called recompression, found a novel way to decide (algorithmically) if a system of equations over a free group has a solution. Ciobanu, Diekert, and Elder use this to give another description of the set of solutions to a system of equations over a free group and show it is an EDTOL language, which is a type of indexed language.

31 A sequence f i : G (Γ; P 1,... P p ) is convergent if for every g G.

32 A sequence f i : G (Γ; P 1,... P p ) is convergent if for every g G. 1. There is some N(g) such that for all either: for all i N(g), f i (g) = Γ 1, or for all i N(g), f i (g) Γ 1.

33 A sequence f i : G (Γ; P 1,... P p ) is convergent if for every g G. 1. There is some N(g) such that for all either: for all i N(g), f i (g) = Γ 1, or for all i N(g), f i (g) Γ For every P j there is some M j (g) such that either: for all i M j (g), f i (g) Γ P j, or for all i M j (g), f i (g) Γ P j.

34 A limit group is: L = G/{g G f i (g) = Γ 1 for almost all i}

35 A limit group is: L = G/{g G f i (g) = Γ 1 for almost all i} Its parabolic structure is: P = {g L its lift ĝ G is almost always sent to a parabolic}

36 A limit group is: L = G/{g G f i (g) = Γ 1 for almost all i} Its parabolic structure is: P = {g L its lift ĝ G is almost always sent to a parabolic} Inna and I show that P is a union of maximal subgroups and that intersections involving maximal parabolic subgroups and maximal virtually abelian subgroups of L are bounded by some k = k ((Γ; P 1,..., P p )). This required proving things about relatively hyperbolic groups everybody knows is true, but that that were not already proved by Osin.

37 The limit group (L, P) admits a canonical JSJ decomposition, in the sense of Guirardel and Levitt, that shows all possible parabolic and virtually abelian splittings.

38 The limit factoring property π G L f i?h i Γ

39 The limit factoring property π G L f i?h i Γ If Γ is equationally Noetherian, then yes. Conversely (also observed by M. Hull) the existence of such h i (for infinitely many i) implies equationally Noetherian. So Equationally Noetherian Limit factoring property

40 We use the Sela s shortening with the Reinfeldt-Weidmann adaptation for hyperbolic groups with torsion. This argument involves action on R-trees. Almost superstable actions.

41 Let G = x 1,..., x n = X and let S be some Γ-space. Then f i : G Γ induce an action on S. ξ i

42 Let G = x 1,..., x n = X and let S be some Γ-space. Then f i : G Γ induce an action on S. ξ i f i (x 1 ) ξ i

43 Let G = x 1,..., x n = X and let S be some Γ-space. Then f i : G Γ induce an action on S. f i (x 2 ) ξ i ξ i f i (x 1 ) ξ i

44 Let G = x 1,..., x n = X and let S be some Γ-space. Then f i : G Γ induce an action on S. f i (x 2 ) ξ i ξ i max disp ξi (f i (X )) f i (x m ) ξ i f i (x 1 ) ξ i

45 Let G = x 1,..., x n = X and let S be some Γ-space. Then f i : G Γ induce an action on S. f i (x 2 ) ξ i ξ i max disp ξi (f i (X )) f i (x m ) ξ i f i (x 1 ) ξ i ξ i is minimally displaced by f i (X ) if it realizes the minimum. This minimum is called the f i (X )-displacement µ i.

46 If (C, d) is a δ-hyperbolic metric space, then we can scale down the metric to get (C, δ µ i ), a δ µ i -hyperbolic metric space.

47 If (C, d) is a δ-hyperbolic metric space, then we can scale down the metric to get (C, δ µ i ), a δ µ i -hyperbolic metric space. If the µ i then we get a (non-trivial) action of G, in fact of L, on an R-tree.

48 Because the cusped space is proper (finite diameter balls are finite)... Lemma (Alibegovic) If the f i don t map G to a parabolic P i Γ and if the maps aren t all Γ-conjugate, then, considering the induced action of G on the cusped space, (for a subsequence) µ i. Good: The f i s give an action on an R-tree T.

49 Because the cusped space is proper (finite diameter balls are finite)... Lemma (Alibegovic) If the f i don t map G to a parabolic P i Γ and if the maps aren t all Γ-conjugate, then, considering the induced action of G on the cusped space, (for a subsequence) µ i. Good: The f i s give an action on an R-tree T. Bad: We have little control on the asymptotic cones of the horoballs, so this action can be really weird. (See Drutu-Sapir.)

50 Maybe horoballs aren t that great. (They are, though.)

51 Maybe horoballs aren t that great. (They are, though.)

52 As for the coned space, horoballs become points in R-trees so... Good: If the µ i then we have an almost superstable action of L on the limiting R-tree. (Note for hipsters: This is actually easy to see for any acylindrically hyperbolic action.)

53 As for the coned space, horoballs become points in R-trees so... Good: If the µ i then we have an almost superstable action of L on the limiting R-tree. (Note for hipsters: This is actually easy to see for any acylindrically hyperbolic action.) Bad: Because the coned space isn t proper (i.e. some finite diameter balls are infinite), even if the f i are non-conjugate there is no reason for µ i.

54 To compromise we must crush.

55 To compromise we must crush. Fact: The hyperbolicity constant is nicely controlled.

56 And pass to parabolic edge free components.

57 And pass to parabolic edge free components

58 Every such component is of the form K = Z; P 1,..., P r, where Z is finite and the P r come from the incoming parabolic edge groups. There is no a priori reason for the P i to be finitely generated. But they will always fix points in asymptotic cones of coned spaces. Furthermore K is one-ended relative to P 1,..., P r. Thankfully Guirardel s paper on superstable actions on R-trees has the necessary results for relatively finitely generated groups.

59 So although on one hand for (almost) all f i K when acting on the coned space we have µ i = N. E.g. N = 10 On the other hand by the Alibegovic lemma, if the f i K are nonconjugate then µ i.

60 So although on one hand for (almost) all f i K when acting on the coned space we have µ i = N. E.g. N = 10 On the other hand by the Alibegovic lemma, if the f i K are nonconjugate then µ i. Inflate or uncrush horoballs and rescale. This must be done with some care to preserve the fact that paths only visit finitely many horoballs...

61 So although on one hand for (almost) all f i K when acting on the coned space we have µ i = N. E.g. N = 10 On the other hand by the Alibegovic lemma, if the f i K are nonconjugate then µ i. Inflate or uncrush horoballs and rescale. This must be done with some care to preserve the fact that paths only visit finitely many horoballs...

62 So although on one hand for (almost) all f i K when acting on the coned space we have µ i = N. E.g. N = 10 On the other hand by the Alibegovic lemma, if the f i K are nonconjugate then µ i. Inflate or uncrush horoballs and rescale. This must be done with some care to preserve the fact that paths only visit finitely many horoballs...

63 We have a tree-graded asymptotic cone.

64 We have a tree-graded asymptotic cone.

65 We have a tree-graded asymptotic cone. With the property that for the distinguished basepoint ξ, for every g L the path [ξ, g ξ] is contained in finitely many pieces.

66 This finiteness property implies that when we dualize we get a simplicial tree.

67 This finiteness property implies that when we dualize we get a simplicial tree.

68 This finiteness property implies that when we dualize we get a simplicial tree.

69 This finiteness property implies that when we dualize we get a simplicial tree. This is a K-tree with parabolic edge groups. I.e. the parabolic edge free component K admits a parabolic splitting, but this is forbidden by the JSJ.

70 So it goes like this. Let K be a parabolic edge free component of L and let it act on the coned space via f i K. 1. If µ i we can use the shortening argument (which involves those automorphisms.) 2. If the µ i are bounded then the f i K must eventually all be conjugate; since otherwise we showed that we can produce a parabolic splitting.

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