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

• f 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

• f 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

• f 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 Sg of genus g ≥ 2 has a

• ne-relator presentation of the form

Sg = < a1, b1, ..., ag, bg; [a1, b1]...[ag, bg] = 1 > while a non-orientable surface group Tg of genus g ≥ 2 also has a one-relator presentation - now of the form Tg = < a1, a2, ..., ag; a2

1a2 2...a2 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 Sg of genus g ≥ 2 and a nonorientable surface group Tg 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

• therwise 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

• therwise very difficult

The solution to the Tarksi Problems implies that any first

• rder 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∃g1, ..., gt, h1, ..., hk} (g−1Rg = g−1

1 S±1g1...g−1 t

S±1gt)∧(g−1Sg = h−1

1 R±1h1...h−1 k R±1hk

= ⇒ {∃x ∈ G(x−1Rx = S ∨ x−1Rx = 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

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 Faithful Linear Representations Theorem Any surface group of genus g ≥ 2 has a faithful representation in PSL(2, ℂ), in fact within PSL(2, ℝ).

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 Faithful Linear Representations Theorem Any surface group of genus g ≥ 2 has a faithful representation in PSL(2, ℂ), in fact within PSL(2, ℝ). One-Relator Presentations All surface groups have

• ne-relator presentations of a specific type.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Some Properties of Surface Groups

Subgroup Properties Let G be a surface group of genus g ≥ 2. Then a subgroup H of finite index greater than 1 is another surface group of higher genus and a subgroup of infinite index must be a free group

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Some Properties of Surface Groups

Subgroup Properties Let G be a surface group of genus g ≥ 2. Then a subgroup H of finite index greater than 1 is another surface group of higher genus and a subgroup of infinite index must be a free group Elementary Free Theorem An orientable surface group of genus g ≥ 2 and a nonorientable surface group of genus g ≥ 4 ie elementary free

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Cyclically Pinched and Conjugacy Pinched Constructions

From the viewpoint of group presentations the algebraic generalization of the one-relator presentation type of a surface group presentation leads to cyclically pinched one-relator groups and conjugacy pinched one-relator groups

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Cyclically Pinched and Conjugacy Pinched Constructions

From the viewpoint of group presentations the algebraic generalization of the one-relator presentation type of a surface group presentation leads to cyclically pinched one-relator groups and conjugacy pinched one-relator groups These groups have the same general form as a surface group and have proved to be quite amenable to study.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Cyclically Pinched and Conjugacy Pinched Constructions

From the viewpoint of group presentations the algebraic generalization of the one-relator presentation type of a surface group presentation leads to cyclically pinched one-relator groups and conjugacy pinched one-relator groups These groups have the same general form as a surface group and have proved to be quite amenable to study. Surprisingly the satisfy many of the same linear properties as surface groups.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Cyclically Pinched One Relator Groups

A cyclically pinched one-relator group is a one-relator group of the following form G = < a1, ..., ap, ap+1, ..., an; U = V > where 1 ∕= U = U(a1, ..., ap) is a cyclically reduced, non-primitive (not part of a free basis) word in the free group F1 on a1, ..., ap and 1 ∕= V = V (ap+1, ..., an) is a cyclically reduced, non-primitive word in the free group F2 on ap+1, ..., an.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Cyclically Pinched One Relator Groups

A cyclically pinched one-relator group is a one-relator group of the following form G = < a1, ..., ap, ap+1, ..., an; U = V > where 1 ∕= U = U(a1, ..., ap) is a cyclically reduced, non-primitive (not part of a free basis) word in the free group F1 on a1, ..., ap and 1 ∕= V = V (ap+1, ..., an) is a cyclically reduced, non-primitive word in the free group F2 on ap+1, ..., an. Such a group is the free product of the free groups on a1, ..., ap and ap+1, ..., an respectively amalgamated over the cyclic subgroups generated by U and V .

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Cyclically Pinched One Relator Groups

Cyclically pinched one-relator groups have been shown to be extremely similar to surface groups. We summarize many of the most important results. Theorem Let G be a cyclically pinched one-relator group. Then (1) G is residually finite (G.Baumslag [GB 1]) (2) G has a solvable conjugacy problem (S.Lipschutz [Li]) and is conjugacy separable (J.Dyer[D]) (3) G is subgroup separable (Brunner,Burns and Solitar[BBS])

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Cyclically Pinched One Relator Groups

Theorem Let G be a cyclically pinched one-relator group. Then (4) If neither U nor V is a proper power then G has a faithful representation over some commutative field (Wehrfriztz[W]). (5) If neither U nor V is a proper power then G has a faithful representation in PSL2(ℂ) (Fine-Rosenberger[FR]) (6) If neither U nor V is a proper power then G is hyperbolic ([BeF],[JR],[KhM]) (7) If neither U nor V is in the commutator subgroup of its respective factor then G is free-by-cyclic (BFMT)

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Cyclically Pinched One Relator Groups

Rosenberger using Nielsen cancellation, has given a positive solution to the isomorphism problem for cyclically pinched

• ne-relator groups, that is, he has given an algorithm to

determine if an arbitrary one-relator group is isomorphic or not to a given cyclically pinched one-relator group.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Conjugacy Pinched One Relator Groups

The HNN analogs of cyclically pinched one-relator groups are called conjugacy pinched one-relator groups and are also motivated by the structure of orientable surface groups. A conjugacy pinched one-relator group is a one-relator group

• f the form

G = < a1, ..., an, t; tUt−1 = V > where 1 ∕= U = U(a1, ..., an) and 1 ∕= V = V (a1, ..., an) are cyclically reduced in the free group F on a1, ..., an.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Conjugacy Pinched One Relator Groups

The HNN analogs of cyclically pinched one-relator groups are called conjugacy pinched one-relator groups and are also motivated by the structure of orientable surface groups. A conjugacy pinched one-relator group is a one-relator group

• f the form

G = < a1, ..., an, t; tUt−1 = V > where 1 ∕= U = U(a1, ..., an) and 1 ∕= V = V (a1, ..., an) are cyclically reduced in the free group F on a1, ..., an. Structurally such a group is an HNN extension of the free group F on a1, ..., an with cyclic associated subgroups generated by U and V and is hence the HNN analog of a cyclically pinched one-relator group.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Conjugacy Pinched One Relator Groups

An extremely important conjugacy pinched construction is an extension of centralizers. Let B be a CSA group. Let U ∈ B not a proper power then the rank one extension of centralizer is the conjugacy pinched construction G =< t, B; relB, t−1Ut >

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Conjugacy Pinched One Relator Groups

An extremely important conjugacy pinched construction is an extension of centralizers. Let B be a CSA group. Let U ∈ B not a proper power then the rank one extension of centralizer is the conjugacy pinched construction G =< t, B; relB, t−1Ut > It has been proved that a finitely generated fully residually free group is embeddable as a subgroup in an iterated extension of centralizers starting with free groups.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Baumslag Doubles

An important subclass of the class of cyclically pinched

• ne-relator groups are the Baumslag doubles

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Baumslag Doubles

An important subclass of the class of cyclically pinched

• ne-relator groups are the Baumslag doubles

Here let F be a free group and U ∈ F. Let F be an identical free group with U the element corresponding to U. Then the amalgamated product F ★U=U F is a Baumslag double.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Representations of Limit Groups

A group G is fully residually free if given any finite set of elements g1, ..., gn ∈ G there exists a homomorphism 휙 : G → F with F a free group such that 휙(gi) ∕= 1 for all i = 1, .., n. If G is finitely generated it is called a limit group. Limit groups played a prominent role in the solution of the Tarksi problems by Kharlampovich and Myasnikov and Sela. In Selas’ approach they arise as limit groups of homomorphisms into free group whence the name.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Representations of Limit Groups

A group G is fully residually free if given any finite set of elements g1, ..., gn ∈ G there exists a homomorphism 휙 : G → F with F a free group such that 휙(gi) ∕= 1 for all i = 1, .., n. If G is finitely generated it is called a limit group. Limit groups played a prominent role in the solution of the Tarksi problems by Kharlampovich and Myasnikov and Sela. In Selas’ approach they arise as limit groups of homomorphisms into free group whence the name. The primary example of a nonfree limit group is an orientable surface group Sg. Originally this was proved to be residually free by G. Baumslag using what is now called the big powers

• argument. As a result of these groups being commutative

transitive, it follows from a result of B.Baumslag, that they are fully residually free.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Faithful Representations of Limit Groups

Using the techniques we used to prove that a cyclically pinched one-relator group has a faithful representation ino PSL(2, C) we were able to prove the following Theorem Let G be a hyperbolic limit group. Then G has a constructive faithful representation 휌 : G → PSL(2, ℂ).

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Faithful Representations of Limit Groups

Using the techniques we used to prove that a cyclically pinched one-relator group has a faithful representation ino PSL(2, C) we were able to prove the following Theorem Let G be a hyperbolic limit group. Then G has a constructive faithful representation 휌 : G → PSL(2, ℂ). Using nonstandard free groups this was extended Theorem Let G be any limit group. Then G has a faithful representation 휌 : G → PSL(2, ℂ).

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Faithful Representations of Limit Groups

The proofs are based on a series of results on faithful representations of certain group amalgams and HNN extensions in PSL(2, ℂ) combined with the structure theory of limit groups.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Faithful Representations of Limit Groups

The proofs are based on a series of results on faithful representations of certain group amalgams and HNN extensions in PSL(2, ℂ) combined with the structure theory of limit groups. These theorems are tied to much older problems related to representations of surface groups and more generally Fuchsian groups into Lie Groups.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Real Representations of Limit Groups

We were asked if these results could be extended to real

• representations. We attempted to prove using the amalgam

techniques that limit groups have faithful representations in PSL(2, ℝ). To do this we needed that that the factors are

• free. Hence it is still open whether an arbitrary limit group

has a faithful 2-dimensional real representation. However Fine,Kreuzer and Rosenberger were able to prove this is true for hyperbolic cyclically pinched one relator groups

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

Real Representations of Limit Groups

Theorem Let G =< a1, ..., an, b1, ..., bm; u = v > with n ≥ 2, m ≥ 2 and u = u(a1, ..., an) is a nontrivial, not primitive and not a proper power in the free group F1 =< a1, ..., an > and v = v(b1, ..., bm) is a nontrivial, not primitive and not a proper power in the free group F2 =< b1, ..., bm >. Then there exists a faithful representation 휙 : G → PSL(2, R)

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

The surface group conjecture as formulated by Melnikov in the Kourovka Notebook is whether a residually finite

• ne-relator group in which every subgroup of finite index ia

again a one-relator group must be a surface group.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

The surface group conjecture as formulated by Melnikov in the Kourovka Notebook is whether a residually finite

• ne-relator group in which every subgroup of finite index ia

again a one-relator group must be a surface group. This original question was false - the residually finite Baumslag-Solitar groups provide counterexamples so the conjecture was modified to Surface Group Conjecture Let G be a one-relator group with the property that any subgroup of finite index is again a

• ne-relator group and any subgroup of infinite index is a free

group, then G must be a surface group.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

There were a collection of partial results concentrating on Property IF - the property that a subgroup of infinite index is free.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

There were a collection of partial results concentrating on Property IF - the property that a subgroup of infinite index is free. Theorem ([FKMRR] Suppose that G is a finitely generated fully residually free group with property IF. Then G is either a free group or a cyclically pinched one relator group or a conjugacy pinched one relator group.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

There were a collection of partial results concentrating on Property IF - the property that a subgroup of infinite index is free. Theorem ([FKMRR] Suppose that G is a finitely generated fully residually free group with property IF. Then G is either a free group or a cyclically pinched one relator group or a conjugacy pinched one relator group. Theorem (FKMRR) Let G be a finitely generated fully residually free group with property IF. Then either G is hyperbolic or G is free abelian of rank 2.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

Based on these partial results several new conjectures were given

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

Based on these partial results several new conjectures were given Surface Group Conjecture A Suppose that G is a residually finite non-free, non-cyclic one-relator group such that every subgroup of finite index is again a one-relator

• group. Then G is either a surface group or a Baumslag-Solitar

group BS(1, m) for some integer m.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

We note that the groups BS(1, 1) and BS(1, −1) are surface

• groups. In surface groups, subgroups of infinite index must be

free groups. To avoid the Baumslag-Solitar groups, BS(1, m), ∣m∣ ≥ 2, Surface Group Conjecture A, was modified to: Surface Group Conjecture B Suppose that G is a non-free, non-cyclic one-relator group such that every subgroup of finite index is again a one-relator group, there exists a noncyclic subgroup of infinite index and and every subgroup of infinite index is a free group . Then G is a surface group (of genus g ≥ 2.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

Surface Group Conjecture C Suppose that G is a finitely generated nonfree freely indecomposable fully residually free group with property IF. Then G is a surface group.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

If we begin in the context of cyclically pinched or conjugacy pinched one-relator groups the following was known Theorem (FKMRR) Let G be a nonfree cyclically pinched or conjugacy pinched

• ne-relator group with property IF. Then each subgroup of finite

index is again a cyclically pinched or conjugacy pinched one-relator group.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

Combining results in[FKMRR] with the recent theorem of H. Wilton below and results of Gildenhuys, Kharlampovich and Myasnikov [GKhM] and Stallings [St], Ciobanu, Fine and Rosenberger can settle conjecture C and completly settle the surface group conjecture in the context of cyclically pinched and conjugacy pinched one-relator groups with property IF. Theorem (Wilton) Let G be a hyperbolic one-ended cyclically pinched

• ne-relator group or a hyperbolic one-ended conjugacy pinched
• ne-relator group. Then either G is a surface group, or G has a

finitely generated non-free subgroup of infinite index.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

We thus have the following:

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

We thus have the following: Theorem Suppose that G is a finitely generated nonfree freely indecomposable fully residually free group with property IF. Then G is a surface group. That is Surface Group Conjecture C is true.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Surface Group Conjecture

Theorem Let G be a cyclically pinched one-relator group amalgamated via u = v where not both u and v are proper powers. Then if G satisfies property IF then G is a surface group. Further suppose that G is a conjugacy pinched one-relator group where < u > and < v > are the associated subgroups, and u and v are not both proper powers. Then if G satisfies property IF then G is a surface group.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Gromov Surface Group Question

A question of Gromov asks whether a one-ended word hyperbolic group must contain a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. Recent work by Gordon and Wilton [GW] and Kim and Wilton [KW] gives sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups

The Gromov Surface Group Question

Using Nielsen cancellation methods we can prove the following: Theorem Let G = F ★

{W =W }

F be a hyperbolic Baumslag double. Then G contains a hyperbolic orientable surface group of genus 2 if and

• nly if W is a commutator, that is W = [U, V ] for some elements

U, V ∈ F. Further a Baumslag double G contains a nonorientable surface group of genus 4 if and only if W = X 2Y 2 for some X, Y ∈ F.

Benjamin Fine - Gerhard Rosenberger On Cyclically Pinched and Conjugacy Pinched One-Relator groups