Some model theory of the free group
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1 Some model theory of the free group Université Lyon 1 July 4, 2017
2 1 2 3
3 s - Algebra A group F is free, if it has the universal property (over a subset S F) for the class of groups. Universal property: for every group G and every function f : S G, there exists a unique homomorphism h : F G such that the above diagram commutes; the subset S is called the basis of F; and the cardinality of S is called the rank of F.
4 s - Topology A group F is free, if it is isomorphic to the fundamental group of a bouquet of circles: The fundamental group of a pointed topological space (X, ) is the group of homotopy classes of loops of X that start and end at (where the group law is induced by the composition of loops).
5 s - Geometry A group F is free, if it admits a free action without inversion on a tree (a nonoriented connected graph without cycles): An action (by graph automorphisms) of a group G on a graph G is free, if g.x x for each g G \ {1} and every vertex x G.
6 s - Geometry A group F is free, if it admits a free action without inversion on a tree (a nonoriented connected graph without cycles): An action (by graph automorphisms) of a group G on a graph G is free, if g.x x for each g G \ {1} and every vertex x G. Theorem (Nielsen-Schreier): A subgroup of a free group is a free group.
7 Question (Tarski): Do nonabelian free groups share the same common first-order theory?
8 Question (Tarski): Do nonabelian free groups share the same common first-order theory? Free abelian groups, Z n, of different ranks have different first-order theories; since [Z n : 2Z n ] [Z m : 2Z m ] for m n.
9 Question (Tarski): Do nonabelian free groups share the same common first-order theory? Free abelian groups, Z n, of different ranks have different first-order theories; since [Z n : 2Z n ] [Z m : 2Z m ] for m n. Question (Malcev): Suppose F n is a free group of rank n. Is the derived subgroup [F n, F n ] definable in F n? Remark: the quotient group F n /[F n, F n ] is isomorphic to Z n.
10 Tarski s Problem Theorem (Sela 2001 / Kharlampovich-Miasnikov): Nonabelian free groups share the same common first-order theory. As a matter of fact the following chain is elementary: F 2 F 3... F n...
11 Tarski s Problem Theorem (Sela 2001 / Kharlampovich-Miasnikov): Nonabelian free groups share the same common first-order theory. As a matter of fact the following chain is elementary: F 2 F 3... F n...
12 In addition, Sela described all finitely generated models of the first-order theory of the free group; he called them Hyperbolic Towers.
13 In addition, Sela described all finitely generated models of the first-order theory of the free group; he called them Hyperbolic Towers.
14 First model theoretic results by Sela Theorem: The theory of the free group is nonequational.
15 First model theoretic results by Sela Theorem: The theory of the free group is nonequational. Theorem: The theory of the free group is stable.
16 First model theoretic results by Sela Theorem: The theory of the free group is nonequational. Theorem: The theory of the free group is stable. Theorem: The theory of the free group (weakly) eliminates imaginaries up to adding some reasonable sorts.
17 1 2 3
18 Theorem (Poizat): F ω is not superstable. Theorem (Poizat): F ω is connected.
19 Theorem (Poizat): F ω is not superstable. Theorem (Poizat): F ω is connected. Theorem (Pillay): An element of a nonabelian free group is generic if and only if it is primitive, i.e. it is part of some basis. Any maximal independent set of realizations of the generic type in F n is a basis of F n.
20 Theorem (Poizat): F ω is not superstable. Theorem (Poizat): F ω is connected. Theorem (Pillay): An element of a nonabelian free group is generic if and only if it is primitive, i.e. it is part of some basis. Any maximal independent set of realizations of the generic type in F n is a basis of F n. Theorem (Pillay / S.): The generic type has infinite weight.
21 Theorem (Louder-Perin-S.): There exists a finitely generated group G = T fg and two (finite) maximal independent sequences of realizations of the generic type in G of different length. Theorem (Brück): For every n < ω, there exists a finitely generated group G n = T fg and two (finite) maximal independent sequences of realizations of the generic type in G n for which the ratio of their lengths is greater than n.
22 Arbitrarily Large Weight
23
24 Homogeneity Theorem (Perin-S. / Ould Houcine): F n is homogeneous. As a matter of fact every nonabelian free group is strongly ℵ 0 -homogeneous.
25 Homogeneity Theorem (Perin-S. / Ould Houcine): F n is homogeneous. As a matter of fact every nonabelian free group is strongly ℵ 0 -homogeneous. Theorem (S.): Each uncountable free group is not ℵ 1 -homogeneous.
26 Most of the surface groups are not homogeneous. Theorem (Dehn-Nielsen-Baer): Aut(π 1 (Σ)) = Homeo(Σ)
27 Forking Independence Theorem (Perin-S.): Let F be a nonabelian free group and b, c F. Then b is independent from c over if and only if F admits a free splitting as B C with b B and c C.
28 Forking Independence Theorem (Perin-S.): Let F be a nonabelian free group and b, c F. Then b is independent from c over if and only if F admits a free splitting as B C with b B and c C. Theorem (Perin-S.): Let F be a nonabelian free group and b, c, A F. Then b is independent from c over A if and only if
29 Ample Hierarchy Theorem (Pillay): The free group is not CM-trivial, i.e. it is 2-ample.
30 Ample Hierarchy Theorem (Pillay): The free group is not CM-trivial, i.e. it is 2-ample. Theorem (Ould Houcine-Tent / S.): The free group is n ample for all n < ω.
31 Ample Hierarchy Theorem (Pillay): The free group is not CM-trivial, i.e. it is 2-ample. Theorem (Ould Houcine-Tent / S.): The free group is n ample for all n < ω. Remark: the main tool for confirming the algebraic conditions of ampleness is Thurston s pseudo-anosov homeomorphisms.
32 Theorem (Byron-S. / S.): No infinite field is interpretable in the free group.
33 Theorem (Byron-S. / S.): No infinite field is interpretable in the free group. Theorem: Let X be a definable set in a nonabelian free group F. Then either X is internal to a finite set of centralizers (of nontrivial elements) or it cannot be given definably the structure of an abelian group.
34 Theorem (Byron-S. / S.): No infinite field is interpretable in the free group. Theorem: Let X be a definable set in a nonabelian free group F. Then either X is internal to a finite set of centralizers (of nontrivial elements) or it cannot be given definably the structure of an abelian group. Theorem (Perin / Byron-S.): Centralizers of elements in nonabelian free groups are pure groups, i.e. the induced structure on a centralizer can be defined by multiplication alone. Remark: this is the first example of a stable group which is ample but no infinite field is interpretable in it.
35 Theorem (S.): The free group has nfcp.
36 Theorem (S.): The free group has nfcp. Theorem (Sela): The free group is nonequational.
37 Theorem (S.): The free group has nfcp. Theorem (Sela): The free group is nonequational. Theorem (Müller-S.): No free product, except Z 2 Z 2, is equational.
38 Theorem (S.): The free group has nfcp. Theorem (Sela): The free group is nonequational. Theorem (Müller-S.): No free product, except Z 2 Z 2, is equational. Theorem (Sela): Any free product of stable groups is stable.
39 1 2 3
40 Definable Groups Question (Malcev): Suppose F n is a free group of rank n. Is the derived subgroup [F n, F n ] definable in F n?
41 Definable Groups Question (Malcev): Suppose F n is a free group of rank n. Is the derived subgroup [F n, F n ] definable in F n? Theorem (Perin-Pillay-S.-Tent / Kharlampovich-Miasnikov / Bestvina-Feighn): Any proper definable subgroup of a nonabelian free group is cyclic.
42 Definable Groups Question (Malcev): Suppose F n is a free group of rank n. Is the derived subgroup [F n, F n ] definable in F n? Theorem (Perin-Pillay-S.-Tent / Kharlampovich-Miasnikov / Bestvina-Feighn): Any proper definable subgroup of a nonabelian free group is cyclic. Conjecture: The only definable groups in the free group are the obvious ones.
43 Finite Index Subgroups Theorem (Sela): Let G be a finitely generated model of the free group. Then G is a hyperbolic tower. Examples: nonabelian free groups, surface groups, π 1 (Σ) with χ(σ) < 1, and free products of these groups.
44 Finite Index Subgroups Theorem (Sela): Let G be a finitely generated model of the free group. Then G is a hyperbolic tower. Examples: nonabelian free groups, surface groups, π 1 (Σ) with χ(σ) < 1, and free products of these groups. Fact: Let G be a free product of nonabelian groups and surface groups π 1 (Σ) (with χ(σ) < 1). Then any finite index subgroup of G is elementarily equivalent to G.
45 Finite Index Subgroups Theorem (Sela): Let G be a finitely generated model of the free group. Then G is a hyperbolic tower. Examples: nonabelian free groups, surface groups, π 1 (Σ) with χ(σ) < 1, and free products of these groups. Fact: Let G be a free product of nonabelian groups and surface groups π 1 (Σ) (with χ(σ) < 1). Then any finite index subgroup of G is elementarily equivalent to G. Theorem (Guirardel-Levitt-S.): Let G be a finitely generated model of the free group. Then either it is the free product of free groups and surface groups, or it has infinitely many subgroups of finite index pairwise non-elementarily equivalent.
46 Superstable part Theorem (Perin-S.): Let φ(x) be a formula over F n. Suppose φ(f n ) φ(f ω ). Then φ(x) is not superstable.
47 Superstable part Theorem (Perin-S.): Let φ(x) be a formula over F n. Suppose φ(f n ) φ(f ω ). Then φ(x) is not superstable. Conjecture: Let φ(x) be a formula over F n. Then φ(x) is superstable if and only if φ(f n ) = φ(f ω ).
48 East Coast versus West Coast Theory Theorem (Sela / Kharlampovich-Miasnikov): The free group admits quantifier elimination up to boolean combinations of formulas.
49 East Coast versus West Coast Theory Theorem (Sela / Kharlampovich-Miasnikov): The free group admits quantifier elimination up to boolean combinations of formulas. Theorem (Perin / Bestvina-Feighn): The free group is not model complete.
50 East Coast versus West Coast Theory Theorem (Sela / Kharlampovich-Miasnikov): The free group admits quantifier elimination up to boolean combinations of formulas. Theorem (Perin / Bestvina-Feighn): The free group is not model complete. Question: Does the free group admit a model companion?
51 References Z. Sela, Diophantine Geometry over Groups VI: the elementary theory of a free group, Geom. Funct. Anal. (GAFA), 16(3): , Z. Sela, Diophantine Geometry over Groups VIII: stability, Ann. of Math. (2), 177: , C. Perin and R. Sklinos, Homogeneity in the free group, Duke Math. J., 161(13): , C. Perin and R. Sklinos, Forking and JSJ decompositions in the free group, J. Eur. Math. Soc. (JEMS), 18(3): , R. Sklinos, On ampleness and pseudo-anosov homeomorphisms in the free group, Turkish J. Math., 39(1):63 80, R. Sklinos, The free group does not have the finite cover property, to appear in the Israel J. Math., A. Pillay and R. Sklinos, The free group has the dimensional order property, Bull. Lond. Math. Soc., 49(1):89 94, 2017.
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