Rizos Sklinos Historical Remarks Model Theoretic Some model theory of the free group Study of the Free Group Further research Rizos Sklinos Université Lyon 1 July 4, 2017
Rizos Sklinos Historical Remarks Model Historical Remarks Theoretic 1 Study of the Free Group Further research Model Theoretic Study of the Free Group 2 Further research 3
Free Groups - Algebra Rizos Sklinos A group F is free, if it has the universal property (over a Historical subset S ⊂ F ) for the class of groups. Remarks Model Theoretic Study of the Free Group Further research 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 .
Free Groups - Topology Rizos Sklinos A group F is free, if it is isomorphic to the fundamental Historical Remarks group of a bouquet of circles: Model Theoretic Study of the Free Group Further research 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).
Free Groups - Geometry Rizos Sklinos A group F is free, if it admits a free action without inversion on a tree (a nonoriented connected graph Historical Remarks without cycles): Model Theoretic Study of the Free Group Further research 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 .
Free Groups - Geometry Rizos Sklinos A group F is free, if it admits a free action without inversion on a tree (a nonoriented connected graph Historical Remarks without cycles): Model Theoretic Study of the Free Group Further research 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.
Rizos Sklinos Question (Tarski): Historical Do nonabelian free groups share the same common first-order Remarks theory ? Model Theoretic Study of the Free Group Further research
Rizos Sklinos Question (Tarski): Historical Do nonabelian free groups share the same common first-order Remarks theory ? Model Theoretic Study of the Free Group Free abelian groups, Z n , of different ranks have different Further research first-order theories; since [ Z n : 2 Z n ] � = [ Z m : 2 Z m ] for m � = n .
Rizos Sklinos Question (Tarski): Historical Do nonabelian free groups share the same common first-order Remarks theory ? Model Theoretic Study of the Free Group Free abelian groups, Z n , of different ranks have different Further research first-order theories; since [ Z n : 2 Z n ] � = [ Z m : 2 Z 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 .
Tarski’s Problem Rizos Sklinos Theorem (Sela 2001 / Kharlampovich-Miasnikov): Historical Nonabelian free groups share the same common first-order Remarks theory. Model Theoretic Study of the Free Group As a matter of fact the following chain is elementary: Further research F 2 ≤ F 3 ≤ . . . ≤ F n ≤ . . .
Tarski’s Problem Rizos Sklinos Theorem (Sela 2001 / Kharlampovich-Miasnikov): Historical Nonabelian free groups share the same common first-order Remarks theory. Model Theoretic Study of the Free Group As a matter of fact the following chain is elementary: Further research F 2 ≤ F 3 ≤ . . . ≤ F n ≤ . . .
Rizos Sklinos In addition, Sela described all finitely generated models of the first-order theory of the free group; Historical Remarks he called them Hyperbolic Towers . Model Theoretic Study of the Free Group Further research
Rizos Sklinos In addition, Sela described all finitely generated models of the first-order theory of the free group; Historical Remarks he called them Hyperbolic Towers . Model Theoretic Study of the Free Group Further research
First model theoretic results by Sela Rizos Sklinos Historical Theorem: Remarks The theory of the free group is nonequational. Model Theoretic Study of the Free Group Further research
First model theoretic results by Sela Rizos Sklinos Historical Theorem: Remarks The theory of the free group is nonequational. Model Theoretic Study of the Free Group Further Theorem: research The theory of the free group is stable.
First model theoretic results by Sela Rizos Sklinos Historical Theorem: Remarks The theory of the free group is nonequational. Model Theoretic Study of the Free Group Further Theorem: research The theory of the free group is stable. Theorem: The theory of the free group (weakly) eliminates imaginaries up to adding some “reasonable” sorts.
Rizos Sklinos Historical Remarks Model Historical Remarks Theoretic 1 Study of the Free Group Further research Model Theoretic Study of the Free Group 2 Further research 3
Theorem (Poizat): Rizos Sklinos F ω is not superstable. Historical Remarks Model Theorem (Poizat): Theoretic Study of the Free Group F ω is connected. Further research
Theorem (Poizat): Rizos Sklinos F ω is not superstable. Historical Remarks Model Theorem (Poizat): Theoretic Study of the Free Group F ω is connected. Further research 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 (Poizat): Rizos Sklinos F ω is not superstable. Historical Remarks Model Theorem (Poizat): Theoretic Study of the Free Group F ω is connected. Further research 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.
Rizos Sklinos Theorem (Louder-Perin-S.): Historical Remarks There exists a finitely generated group G | = T fg and two Model Theoretic (finite) maximal independent sequences of realizations of the Study of the Free Group generic type in G of different length. Further research 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 .
Arbitrarily Large Weight Rizos Sklinos Historical Remarks Model Theoretic Study of the Free Group Further research
Rizos Sklinos Historical Remarks Model Theoretic Study of the Free Group Further research
Homogeneity Rizos Sklinos Historical Theorem (Perin-S. / Ould Houcine): Remarks F n is homogeneous. Model Theoretic Study of the Free Group Further research As a matter of fact every nonabelian free group is strongly ℵ 0 -homogeneous .
Homogeneity Rizos Sklinos Historical Theorem (Perin-S. / Ould Houcine): Remarks F n is homogeneous. Model Theoretic Study of the Free Group Further research 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.
Rizos Sklinos Most of the surface groups are not homogeneous. Historical Remarks Model Theoretic Study of the Free Group Further research Theorem (Dehn-Nielsen-Baer): Aut ( π 1 (Σ)) ∼ = Homeo (Σ)
Forking Independence Theorem (Perin-S.): Rizos Sklinos Let F be a nonabelian free group and ¯ c ⊂ F . Then ¯ b , ¯ b is Historical Remarks independent from ¯ c over ∅ if and only if F admits a free Model splitting as B ∗ C with ¯ b ⊂ B and ¯ c ⊂ C . Theoretic Study of the Free Group Further research
Forking Independence Theorem (Perin-S.): Rizos Sklinos Let F be a nonabelian free group and ¯ c ⊂ F . Then ¯ b , ¯ b is Historical Remarks independent from ¯ c over ∅ if and only if F admits a free Model splitting as B ∗ C with ¯ b ⊂ B and ¯ c ⊂ C . Theoretic Study of the Free Group Theorem (Perin-S.): Further research Let F be a nonabelian free group and ¯ c , A ⊂ F . Then ¯ b , ¯ b is independent from ¯ c over A if and only if
Ample Hierarchy Theorem (Pillay): Rizos Sklinos The free group is not CM-trivial, i.e. it is 2-ample. Historical Remarks Model Theoretic Study of the Free Group Further research
Ample Hierarchy Theorem (Pillay): Rizos Sklinos The free group is not CM-trivial, i.e. it is 2-ample. Historical Remarks Model Theoretic Theorem (Ould Houcine-Tent / S.): Study of the Free Group The free group is n − ample for all n < ω . Further research
Ample Hierarchy Theorem (Pillay): Rizos Sklinos The free group is not CM-trivial, i.e. it is 2-ample. Historical Remarks Model Theoretic Theorem (Ould Houcine-Tent / S.): Study of the Free Group The free group is n − ample for all n < ω . Further research Remark: the main tool for confirming the algebraic conditions of ampleness is Thurston’s pseudo-Anosov homeomorphisms.
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