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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
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
This talk is based on joint results with A. Myasnikov and D. Serbin.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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 .
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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 .
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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 .
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
Z2 with the right-lex ordering
(0,0)
(0,-1)
(0,1)
x
y
One-dimensional picture
( (( (( (( ((0,0)(0,-1) (0,1)
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
Z2 with the right-lex ordering
(0,0)
(0,-1)
(0,1)
x
y
One-dimensional picture
( (( (( (( ((0,0)(0,-1) (0,1)
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
Generalize Bass-Serre theory (for actions on Z-trees) to actions onarbitrary Λ-trees.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
Examples for Λ = R
X = R with usual metric.
A geometric realization of a simplicial tree.
SNCF system
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
Examples for Λ = R
X = R with usual metric.
A geometric realization of a simplicial tree.
SNCF system
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
Examples for Λ = R
X = R with usual metric.
A geometric realization of a simplicial tree.
SNCF system
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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
x
(x1,y1)
(x2,y2)
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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).
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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).
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination Processes
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).
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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 .
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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 .
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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 ,
where
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 .
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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].
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
G
Ian Agol The virtual Haken conjectureOlga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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).
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
Corollaries
Theorem
Every finitely presented Λ-free group G has a quasi-convexhierarchy.
Theorem
Every finitely presented Λ-free group is locally quasi-convex.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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
g
f
1
h
c(f,g)=
c(f,h)
c(g,h)
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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
g
f
1
h
c(f,g)=
c(f,h)
c(g,h)
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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
g
f
1
h
c(f,g)=
c(f,h)
c(g,h)
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
Free length functions
A length function l : G → Λ is free if l(g2) > l(g) for everynon-trivial g ∈ G .
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees
Finitely presented Λ-free groupsNon-Archimedean Infinite words
Elimination ProcessesLyndon length functions
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.
Olga Kharlampovich (Hunter College, CUNY) Groups acting on Λ trees