Lecture 10: Limit groups

Transcription

1 Lecture 10: Limit groups Olga Kharlampovich November 4 1 / 16

2 Groups universally equivalent to F Unification Theorem 1 Let G be a finitely generated group and F G. Then the following conditions are equivalent: 1) G is discriminated by F (fully residually free), i.e. for any finite subset M G there exists a homomorphism G F injective on M. 2) G is universally equivalent to F ; 3) G is the coordinate group of an irreducible variety over F. 4) G is a Sela s limit group. 5) G is a limit of free groups in Gromov-Hausdorff metric. 6) G embeds into an ultrapower of free groups. 2 / 16

3 Groups universally equivalent to F Unification Theorem 1 Let G be a finitely generated group and F G. Then the following conditions are equivalent: 1) G is discriminated by F (fully residually free), i.e. for any finite subset M G there exists a homomorphism G F injective on M. 2) G is universally equivalent to F ; 3) G is the coordinate group of an irreducible variety over F. 4) G is a Sela s limit group. 5) G is a limit of free groups in Gromov-Hausdorff metric. 6) G embeds into an ultrapower of free groups. 2 / 16

4 Groups universally equivalent to F Unification Theorem 2 (with coefficients) Let G be a finitely generated group containing a free non-abelian group F as a subgroup. Then the following conditions are equivalent: 1) G is F -discriminated by F ; 2) G is universally equivalent to F (in the language with constants); 3) G is the coordinate group of an irreducible variety over F. 4) G is Sela s limit group. 5) G is a limit of free groups in Gromov-Hausdorff metric. 6) G F -embeds into an ultrapower of F. 3 / 16

5 limits of free groups Limit groups by Sela H fg, a sequence {φ i } in Hom(H, F ) is stable if, for all h H, h φ i is eventually always 1 or eventually never 1. Ker φ i is: {h H h φ i = 1 for almost all i} G is a limit group if there is a fg H and a stable sequence {φ i } such that: G = H/Ker φ i. 4 / 16

6 limits of free groups [Ch. Champetier and V. Guirardel (2004)] A marked group (G, S) is a group G with a prescribed family of generators S = (s 1,..., s n ). Two marked groups (G, (s 1,..., s n )) and (G, (s 1,..., s n)) are isomorphic as marked groups if the bijection s i s i extends to an isomorphism. Equivalently, their Cayley graphs are isomorphic as labelled graphs. For example, ( a, (1, a)) and ( a, (a, 1)) are not isomorphic as marked groups. Denote by G n the set of groups marked by n elements up to isomorphism of marked groups. 5 / 16

7 A topology on spaces of marked groups Topology on by G n The topology in terms of normal subgroups The generating set S of (G, S) induces a word metric on G. We denote by B (G,S) (R) its ball of radius R centered at the identity. Let 2 Fn be the set of all subsets of the free group F n. For any subsets A, A consider v(a, A ) = max{r N A B Fn,(s 1,...,s n)(r) = A B Fn,(s 1,...,s n)(r)}. It induces a metric d on 2 Fn defined by d(a, A ) = e v(a,a ). This metric is ultrametric and makes 2 Fn in compact metric space. The topology in terms of epimorphisms Two epimorphisms F n G i, i = 1, 2 of G n are close if their kernels are close. The topology in terms of relations One can define a metric on G n by setting the distance between two marked groups (G, S) and (G, S ) to be e N if they have exactly the same relations of length at most N (under the bijection S S ) (Grigorchuk, Gromov s metric) Finally, a limit group is a limit (with respect to the metric above) 6 / 16

8 limits of free groups Examples: 1) An infinite presented group < s 1,..., s n r 1,..., r i,... > marked by s 1,..., s n is a limit of finitely presented groups < s 1,..., s n r 1,..., r i >, when i. 2) Z/iZ converges to Z. 3) A free abelian group of rank 2 is a limit of a sequence of cyclic groups with marking ( a, (a, a n )), n. 4) HW 13 Prove that a sequence of markings of Z converges to some marking of Z k. 5) F k is a limit of markings of F 2. 6) A residually finite group is a limit of finite groups. 7 / 16

9 limits of free groups In the definition of a limit group, F can be replaced by any equationally Noetherian group or algebra. A direct limit of a direct system of n-generated finite partial subalgebras of A such that all products of generators eventually appear in these partial subalgebras, is called a limit algebra over A. It is interesting to study limits of free semigroups. 8 / 16

10 Lemma Let (G, S) be a marling of a finitely presented groups. There exists a neighborhood of (G, S) containing only marked quotients of (G, S). Corollary A finitely presented group which is a limit of the set of finite groups is residually finite. Problem Describe the closure of the set of finite groups in G n. 9 / 16

11 The equivalence 2) 6) is a particular case of general results in model theory. An ultrafilter on N is a finitely additive measure of total mass 1, defined on all subsets of N, and with values in {0, 1}. In other words, it is a map ω : 2 N {0, 1}, such that for all subsets A, B such that A B =, ω(a B) = ω(a) = ω(b), ω(n) = 1. Definition The ultraproduct with respect to ω of a sequence of groups G k is the group ( k N G k )/ ω. Theorem (Los) Let G be a group and G an ultrapower of G. Then G and G have the same elementary theory. 10 / 16

12 5) 6). It is shown by Christophe Champetier, Vincent Guirardel Limit groups as limits of free groups: compactifying the set of free groups ( that a group is a limit group if and only if it is a finitely generated subgroup of an ultraproduct of free groups (for a non-principal ultrafilter), and any such ultraproduct of free groups contains all the limit groups. This implies the equivalence 5) 6). 11 / 16

13 More properties Proposition 3 Let G be a fully residually free group. Then G possesses the following properties. HW12 Prove statements 1,2,3,4 1 Each Abelian subgroup of G is contained in a unique maximal finitely generated Abelian subgroup, in particular, each Abelian subgroup of G is finitely generated; 2 G is finitely presented, and has only finitely many conjugacy classes of its maximal non-cyclic Abelian subgroups. 3 G has solvable word problem; 4 Every 2-generated subgroup of G is either free or abelian; 5 G is linear; 6 If rank (G)=3 then either G is free of rank 3, free abelian of rank 3, or a free rank one extension of centralizer of a free group of rank 2 (that is G = x, y, t [u(x, y), t] = 1, where the word u is not a proper power). 12 / 16

14 More properties Proof of 5) The ultraproduct of SL 2 (Z) is SL 2 ( Z), where Z is the ultpaproduct of Z. (Indeed, the direct product SL 2 (Z) is isomorphic to SL 2 ( Z). Therefore, one can define a homomorphism from the ultraproduct of SL 2 (Z) onto SL 2 ( Z). Since the intersection of a finite number of sets from an ultrafilter again belongs to the ultrafilter, this epimorphism is a monomorphism.) Being finitely generated G embeds in SL 2 (R), where R is a finitely generated subring in Z. 13 / 16

15 Krull dimension Theorem [announced by Louder] There exists a function g(n) such that the length of every proper descending chain of closed sets in F n is bounded by g(n). 14 / 16

16 Description of solutions Theorem [Razborov] Given a finite system of equations S(X ) = 1 in F (A) there is an algorithm to construct a finite Solution Diagram that describes all solutions of S(X ) = 1 in F. 15 / 16

17 Solution diagrams F R(S) σ 1 F R(Ωv1 ) F R(Ωv2 ) F R(Ωvn ) σ 2 F R(Ωv21 ) F R(Ωv2m ) F (A) F (T ) F (A) 16 / 16