Olga Kharlampovich (Hunter College, CUNY) Paris,...

Finitely presented Λ-free groupsNon-Archimedean Infinite words

Elimination Processes

Groups acting on Λ trees

Olga Kharlampovich(Hunter College, CUNY)

Paris, 2014

Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees

This talk is based on joint results with A. Myasnikov and D. Serbin.

The starting point

Theorem (Bass, Serre, 1980). A group G is free if and only if itacts freely on a tree.

Free action = no inversion of edges and stabilizers of vertices aretrivial.

Ordered abelian groups

Λ = an ordered abelian group (any a, b ∈ Λ are comparable and forany c ∈ Λ : a ≤ b ⇒ a + c ≤ b + c).

Examples:

Archimedean case:

Λ = R, Λ = Z with the usual order.

Non-Archimedean case:

Λ = Z2 with the right lexicographic order:

(a, b) < (c , d) ⇐⇒ b < d or b = d and a < c .

Examples:

Archimedean case:

Examples:

Archimedean case:

Z2 with the right-lex ordering

(0,-1)

One-dimensional picture

( (( (( (( ((0,0)(0,-1) (0,1)

Z2 with the right-lex ordering

(0,-1)

One-dimensional picture

( (( (( (( ((0,0)(0,-1) (0,1)

Λ-trees

Morgan and Shalen (1985) defined Λ-trees:A Λ-tree is a metric space (X , p) (where p : X × X → Λ) whichsatisfies the following properties:

1) (X , p) is geodesic,

2) if two segments of (X , p) intersect in a single point, which isan endpoint of both, then their union is a segment,

3) the intersection of two segments with a common endpoint isalso a segment.

Alperin and Bass (1987) developed the theory of Λ-trees andstated the fundamental research goals:

Find the group theoretic information carried by an action on aΛ-tree.

Λ-trees

Morgan and Shalen (1985) defined Λ-trees:A Λ-tree is a metric space (X , p) (where p : X × X → Λ) whichsatisfies the following properties:

1) (X , p) is geodesic,

2) if two segments of (X , p) intersect in a single point, which isan endpoint of both, then their union is a segment,

3) the intersection of two segments with a common endpoint isalso a segment.

Alperin and Bass (1987) developed the theory of Λ-trees andstated the fundamental research goals:

Find the group theoretic information carried by an action on aΛ-tree.

Generalize Bass-Serre theory (for actions on Z-trees) to actions onarbitrary Λ-trees.

Examples for Λ = R

X = R with usual metric.

A geometric realization of a simplicial tree.

SNCF system

Examples for Λ = R

SNCF system

Examples for Λ = R

SNCF system

Examples for Λ = R

X = R2 with metric d defined by

d((x1, y1), (x2, y2)) =

{|y1|+ |y2|+ |x1 − x2| if x1 6= x2

|y1 − y2| if x1 = x2

(x1,y1)

(x2,y2)

Finitely generated R-free groups

Rips’ Theorem [Rips, 1991 - not published]

A f.g. group acts freely on R-tree if and only if it is a free productof surface groups (except for the non-orientable surfaces of genus1,2, 3) and free abelian groups of finite rank.

Gaboriau, Levitt, Paulin (1994) gave a complete proof of Rips’Theorem.

Bestvina, Feighn (1995) gave another proof of Rips’ Theoremproving a more general result for stable actions on R-trees.

Properties

Some properties of groups acting freely on Λ-trees (Λ-freegroups)

1 The class of Λ-free groups is closed under taking subgroupsand free products.

2 Λ-free groups are torsion-free.

3 Λ-free groups have the CSA-property (maximal abeliansubgroups are malnormal).

4 Commutativity is a transitive relation on the set of non-trivialelements.

5 Any two-generator subgroup of a Λ-free group is either free orfree abelian.

The Fundamental Problem

The following is a principal step in the Alperin-Bass’ program:

Open Problem [Alperin, Bass]

Describe finitely generated (finitely presented) groups acting freelyon Λ-trees.

Here ”describe” means ”describe in the standard group-theoreticterms”.

We solved this problem for finitely presented groups.Λ-free groups = groups acting freely on Λ-trees.

Non-Archimedean actions

Theorem (H.Bass, 1991)

A finitely generated (Λ⊕ Z)-free group is the fundamental groupof a finite graph of groups with properties:

vertex groups are Λ-free,

edge groups are maximal abelian (in the vertex groups),

edge groups embed into Λ.

Since Zn ' Zn−1 ⊕ Z this gives the algebraic structure of Zn-freegroups.

Non-Archimedean actions

Theorem (H.Bass, 1991)

A finitely generated (Λ⊕ Z)-free group is the fundamental groupof a finite graph of groups with properties:

vertex groups are Λ-free,

edge groups are maximal abelian (in the vertex groups),

edge groups embed into Λ.

Since Zn ' Zn−1 ⊕ Z this gives the algebraic structure of Zn-freegroups.

Actions on Rn-trees

Theorem [Guirardel, 2003]

A f.g. freely indecomposable Rn-free group is isomorphic to thefundamental group of a finite graph of groups, where each vertexgroup is f.g. Rn−1-free, and each edge group is cyclic.

However, the converse is not true.

Corollary A f.g. Rn-free group is hyperbolic relative to abeliansubgroups.

Notice, that Zn-free groups are Rn-free.

Actions on Rn-trees

Zn-free groups

Theorem [Kharlampovich, Miasnikov, Remeslennikov, 96]

Finitely generated fully residually free groups (= limit groups) areZn-free.

Theorem [Martino and Rourke, 2005]

Let G1 and G2 be Zn-free groups. Then the amalgamated productG1 ∗C G2 is Zm-free for some m ∈ N, provided C is cyclic andmaximal abelian in both factors.

Zn-free groups

Theorem [Kharlampovich, Miasnikov, Remeslennikov, 96]

Finitely generated fully residually free groups (= limit groups) areZn-free.

Theorem [Martino and Rourke, 2005]

Let G1 and G2 be Zn-free groups. Then the amalgamated productG1 ∗C G2 is Zm-free for some m ∈ N, provided C is cyclic andmaximal abelian in both factors.

Examples of Zn-free groups:

R-free groups,

〈 x1, x2, x3 | x21 x2

2 x23 = 1 〉 is Z2-free (but is neither R-free, nor

fully residually free).

R-free groups,

〈 x1, x2, x3 | x21 x2

R-free groups,

〈 x1, x2, x3 | x21 x2

Elimination ProcessesLyndon length functions

From actions to length functions

Let G be a group acting on a Λ-tree (X , d). Fix a point x0 ∈ Xand consider a function l : G → Λ defined by

l(g) = d(x0 , gx0)

l is called a based length function on G with respect to x0, or aLyndon length function.

l is free if the underlying action is free.

Example. In a free group F , the function f → |f | is a freeZ-valued (Lyndon) length function.

Regular action

Definition

Let G act on a Λ-tree Γ. The action is regular with respect tox ∈ Γ if for any g , h ∈ G there exists f ∈ G such that[x , fx ] = [x , gx ] ∩ [x , hx ].

Comments

Let G act on a Λ-tree (Γ, d). Then the action of G is regular withrespect to x ∈ Γ if and only if the length function lx : G → Λ basedat x is regular.Let G act minimally on a Λ-tree Γ. If the action of G is regularwith respect to x ∈ Γ then all branch points of Γ are G -equivalent.Let G act on a Λ-tree Γ. If the action of G is regular with respectto x ∈ Γ then it is regular with respect to any y ∈ Gx .

Regular action

Definition

Let G act on a Λ-tree Γ. The action is regular with respect tox ∈ Γ if for any g , h ∈ G there exists f ∈ G such that[x , fx ] = [x , gx ] ∩ [x , hx ].

Comments

Let G act on a Λ-tree (Γ, d). Then the action of G is regular withrespect to x ∈ Γ if and only if the length function lx : G → Λ basedat x is regular.Let G act minimally on a Λ-tree Γ. If the action of G is regularwith respect to x ∈ Γ then all branch points of Γ are G -equivalent.Let G act on a Λ-tree Γ. If the action of G is regular with respectto x ∈ Γ then it is regular with respect to any y ∈ Gx .

Main Results

Theorem

Any finitely presented regular Λ-free group G can be represented asa union of a finite series of groups

G1 < G2 < · · · < Gn = G ,

1 G1 is a free group,

2 Gi+1 is obtained from Gi by finitely many HNN-extensions inwhich associated subgroups are maximal abelian, finitelygenerated, and the associated isomorphisms preserve thelength induced from Gi .

Main Results

Theorem

Any finitely presented Λ-free group can be isometrically embeddedinto a finitely presented regular Λ-free group.

Theorem

Any finitely presented Λ-free group G is Rn-free for an appropriaten ∈ N, where Rn is ordered lexicographically.

Main Results

Theorem

Let G be a finitely presented group with a free Lyndon lengthfunction l : G → Λ. Then the subgroup Λ0 generated by l(G ) in Λis finitely generated.

Corollaries

Theorem

Any finitely presented Λ-free group G can be obtained from freegroups by a finite sequence of amalgamated free products andHNN extensions along maximal abelian subgroups, which are freeabelain groups of finite rank.

Notice that every f.g. Rn-free group is f.p.

Corollaries

The following result concerns with abelian subgroups of Λ-freegroups. For Λ = Zn it follows from the main structural result forZn-free groups, for Λ = Rn it was proved by Guirardel. Thestatement 1) below answers to Question 2 (page 250) from [Ch] inthe affirmative for finitely presented Λ-free groups.

Theorem

Let G be a finitely presented Λ-free group. Then:

1) every abelian subgroup of G is a free abelian group of finiterank, which is uniformly bounded from above by the rank ofthe abelianization of G .

2) G has only finitely many conjugacy classes of maximalnon-cyclic abelian subgroups,

3) G has a finite classifying space and the cohomologicaldimension of G is at most max{2, r} where r is the maximalrank of an abelian subgroup of G .Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees

Corollaries

Theorem

Every finitely presented Λ-free group is hyperbolic relative to itsnon-cyclic abelian subgroups.

It follows from our structural Theorem and the CombinationTheorem for relatively hyperbolic groups [Dahmani,2003].

Corollaries

The following results answers affirmatively in the strongest form tothe Problem (GO3) from the Magnus list of open problems in thecase of finitely presented groups.

Corollary

Every finitely presented Λ-free group is biautomatic.

Proof.

It follows the structure theorem and Rebbecchi’s result.

Right-angled Artin groups

Introduction Definitions Virtual fibering Implications Proofs

RAAGs (a la Vogtmann)

Right-angled Artin groups

! = simplicial graphThe right-angled Artin group A! is given by:

Generators: nodes of ! Relators: vw = wv if [v,w] is an edge of !

Examples:

F5 !2 *

F3 !1(M3) F2 x F3 !5

Theorem [Droms] A! is a 3-manifold group if and only if ! is a disjoint union of trees and triangles.

Ian Agol The virtual Haken conjectureOlga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees

Corollaries

Definition

A hierarchy for a group G is a way to repeatedly build it startingwith trivial groups by repeatedly takling amalgamated productsA ∗C B and HNN extensions A∗C t=D whose vertex groups haveshorter length hierarchies. The hierarchy is quasi convex if theamalgamated subgroup C is a finitely generated that embeds by aquasi-isometric embedding, and if C is malnormal in A ∗c B orA∗C t=D .

Theorem

(Wise) Suppose G is total relatively hyperbolic and has amalnormal quasi convex hierarchy. Then G is virtually special(therefore has a finite index subgroup that is a subgroup of aRAAG).

Corollaries

Theorem

Every finitely presented Λ-free group G has a quasi-convexhierarchy.

Theorem

Every finitely presented Λ-free group is locally quasi-convex.

Corollaries

Since a finitely generated Rn-free group G is hyperbolic relative toto its non-cyclic abelian subgroups and G admits a quasi-convexhierarchy then the result of D. Wise imply the following.

Corollary

Every finitely presented Λ-free group G is virtually special, that is,some subgroup of finite index in G embeds into a right-angledArtin group.

Corollaries

In his book Chiswell posted Question 3 (page 250): Is every Λ-freegroup orderable, or at least right-orderable? The following resultanswers in the affirmative to Question 3 (page 250) fromChiswell’s book in the case of finitely presented groups.

Theorem

Every finitely presented Λ-free group is right orderable.

Corollaries

The following addresses Chiswell’s question whether Λ-free groupsare orderable or not.

Theorem

Every finitely presented Λ-free group is virtually orderable, that is,it contains an orderable subgroup of finite index.

Since right-angled Artin groups are linear and the class of lineargroups is closed under finite extension we get the following

Theorem

Every finitely presented Λ-free group is linear.

Corollaries

Since every linear group is residually finite we get the following.

Corollary

Every finitely presented Λ-free group is residually finite.

Corollary

Let G be a finitely presented regular Λ-free group. Then thefollowing algorithmic problems are decidable in G :

the Word and Conjugacy Problems;,

the Diophantine Problem (decidability of arbitrary equationsin G ).

Indeed, decidability of equations follows from Dahmani.

Corollaries

Results of Dahmani and Groves imply the following two corollaries.

Corollary

Let G be a finitely presented regular Λ-free group. Then:

G has a non-trivial abelian splitting and one can find such asplitting effectively,

G has a non-trivial abelian JSJ-decomposition and one canfind such a decomposition effectively.

Corollary

The isomorphism problem is decidable in the class of finitelypresented Λ-free groups.The Subgroup Membership Problem is decidable in every finitelypresented Λ-free group.

Length functions

Length functions were introduced by Lyndon (1963).

Let G be a group. A function l : G → Λ is called a length functionon G if

(L1) ∀ g ∈ G : l(g) > 0 and l(1) = 0,

(L2) ∀ g ∈ G : l(g) = l(g−1),

(L3) the triple {c(g , f ), c(g , h), c(f , h)} is isosceles for allg , f , h ∈ G , where c(f , g) is the Gromov’s product:

c(g , f ) =1

2(l(g) + l(f )− l(g−1f )).

{a, b, c} is isosceles = at least two of a, b, c are equal, and notgreater than the third.

Length functions

(L1) ∀ g ∈ G : l(g) > 0 and l(1) = 0,

(L2) ∀ g ∈ G : l(g) = l(g−1),

c(g , f ) =1

2(l(g) + l(f )− l(g−1f )).

Length functions

(L1) ∀ g ∈ G : l(g) > 0 and l(1) = 0,

(L2) ∀ g ∈ G : l(g) = l(g−1),

c(g , f ) =1

2(l(g) + l(f )− l(g−1f )).

Examples of length functions

In a free group F , the function f → |f | is a Z-valued lengthfunction.

For f , g , h ∈ F we have

c(f,g)=

c(f,h)

c(g,h)

c(f,g)=

c(f,h)

c(g,h)

c(f,g)=

c(f,h)

c(g,h)

Free length functions

A length function l : G → Λ is free if l(g2) > l(g) for everynon-trivial g ∈ G .

Chiswell’s Theorem

Theorem [Chiswell]

Let L : G → Λ be a Lyndon length function on a group G . Thenthere exists a Λ-tree (X , d), x ∈ X , and an isometric action of Gon X such that L(g) = d(x , gx) for all g ∈ G .

Notice that L(g) = d(x , gx) is free iff the action of G on X is free.

This gives another approach to free actions on Λ-trees.

Theorem [Chiswell]

Olga Kharlampovich (Hunter College, CUNY) Paris,...

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