On Cyclically Pinched and Conjugacy Pinched One-Relator groups - PowerPoint PPT Presentation

On Cyclically Pinched and Conjugacy Pinched One-Relator groups Benjamin Fine - Gerhard Rosenberger May 28, 2013 Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Surface Groups Surface groups have played a pivotal role in the development of combinatorial group theory and in more recent innovations such as the algebraic geometry over groups Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Surface Groups Surface groups have played a pivotal role in the development of combinatorial group theory and in more recent innovations such as the algebraic geometry over groups From the standpoint of presentations the natural algebraic generalization of surface groups are cyclically pinched and conjugacy pinched one-relator groups Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Surface Groups Surface groups have played a pivotal role in the development of combinatorial group theory and in more recent innovations such as the algebraic geometry over groups From the standpoint of presentations the natural algebraic generalization of surface groups are cyclically pinched and conjugacy pinched one-relator groups In this talk we will review some basic results on these constructions and then talk about some new results. Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Table of Contents Surface Groups Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Table of Contents Surface Groups Tarski Problems and Elementary Free groups Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Table of Contents Surface Groups Tarski Problems and Elementary Free groups Some Properties of Surface Groups Cyclically Pinched and Conjugacy Pinched One-Relator Groups Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Table of Contents Constructive Faithful Representations of Limit groups Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Table of Contents Constructive Faithful Representations of Limit groups Faithful Real Reps of Pinched Groups Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Table of Contents Constructive Faithful Representations of Limit groups Faithful Real Reps of Pinched Groups The Surface group Conjecture Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Table of Contents Constructive Faithful Representations of Limit groups Faithful Real Reps of Pinched Groups The Surface group Conjecture Gromov’s Surface Group Question Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Surface Groups Recall that a surface group is the fundamental group of a compact orientable or non-orientable surface. If the genus of the surface is g then we say that the corresponding surface group also has genus g . Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Surface Groups Recall that a surface group is the fundamental group of a compact orientable or non-orientable surface. If the genus of the surface is g then we say that the corresponding surface group also has genus g . An orientable surface group S g of genus g ≥ 2 has a one-relator presentation of the form S g = < a 1 , b 1 , ..., a g , b g ; [ a 1 , b 1 ] ... [ a g , b g ] = 1 > while a non-orientable surface group T g of genus g ≥ 2 also has a one-relator presentation - now of the form T g = < a 1 , a 2 , ..., a g ; a 2 1 a 2 2 ... a 2 g = 1 > . Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Surface Groups Much of combinatorial group theory arose originally out of the theory of one-relator groups and the concepts and ideas surrounding the Freiheitssatz or Independence Theorem of Magnus. Going backwards the ideas of the Freiheitssatz were motivated by the topological properties of surface groups. Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Surface Groups Much of combinatorial group theory arose originally out of the theory of one-relator groups and the concepts and ideas surrounding the Freiheitssatz or Independence Theorem of Magnus. Going backwards the ideas of the Freiheitssatz were motivated by the topological properties of surface groups. In the structure theory of limit groups surface groups play a prominent role as the prototype example of an elementary free group that is a finitely generated group having the same elementary theory as a free group. Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Tarski Problems and Elementary Free Groups The solution to the Tarksi Problems says that all nonabelian free groups have the same first-order or elementary theory. It was asked prior whether there are nonfree groups with the same elementary theory as the nonabelian free groups Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Tarski Problems and Elementary Free Groups The solution to the Tarksi Problems says that all nonabelian free groups have the same first-order or elementary theory. It was asked prior whether there are nonfree groups with the same elementary theory as the nonabelian free groups An elementary free group or elementarily free group is a group that has the same elementary theory as the class of nonelementary free group. Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Tarski Problems and Elementary Free Groups The solution to the Tarksi Problems says that all nonabelian free groups have the same first-order or elementary theory. It was asked prior whether there are nonfree groups with the same elementary theory as the nonabelian free groups An elementary free group or elementarily free group is a group that has the same elementary theory as the class of nonelementary free group. Theorem An orientable surface group S g of genus g ≥ 2 and a nonorientable surface group T g of genus g ≥ 4 are elementary free. Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Tarski Problems and Elementary Free Groups The fact that the surface groups are elementary free provides a powerful tool to prove things in surface groups that are otherwise very difficult Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Tarski Problems and Elementary Free Groups The fact that the surface groups are elementary free provides a powerful tool to prove things in surface groups that are otherwise very difficult The solution to the Tarksi Problems implies that any first order theorem holding in the class of nonabelian free groups must also hold in surface groups. In many cases proving these results directly is very nontrivial. Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Tarski Problems and Elementary Free Groups Magnus proved the following theorem about the normal closures of elements in nonabelian free groups: Theorem Let F be a nonabelian free group and R , S ∈ F. Then if N ( R ) = N ( S ) it follows that R is conjugate to either S or S − 1 . Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Tarski Problems and Elementary Free Groups Magnus proved the following theorem about the normal closures of elements in nonabelian free groups: Theorem Let F be a nonabelian free group and R , S ∈ F. Then if N ( R ) = N ( S ) it follows that R is conjugate to either S or S − 1 . J. Howie and independently O. Bogopolski gave a proof of this for surface groups. Their proofs were nontrivial. However with a bit of work it can be determined that this is actually a first order theorem and hence from the Tarksi problems it holds automatically in surface groups. Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Tarski Problems and Elementary Free Groups A sequence of elementary sentences of the form {∀ R , S ∈ G , ∀ g ∈ , G ∃ g 1 , ..., g t , h 1 , ..., h k } ( g − 1 Rg = g − 1 1 S ± 1 g 1 ... g − 1 S ± 1 g t ) ∧ ( g − 1 Sg = h − 1 1 R ± 1 h 1 ... h − 1 k R ± 1 h k t ⇒ {∃ x ∈ G ( x − 1 Rx = S ∨ x − 1 Rx = S − 1 ) } = Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Some Properties of Surface Groups Surface groups of course have many properties which in turn have sparked the study of these properties in general groups. For this talk we concentrate on three and their generalizations to cyclically pinched groups Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups