Download Equations in groups, actions on trees, algorithmic

Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Equations in groups, actions on trees, algorithmic
topology and dynamical systems.
Olga Kharlampovich
(McGill University)
Newcastle, 2010
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Equations in groups, Elimination process
I will talk about an (infinite) rewriting process of a certain type
that transforms formal systems of equations in groups or
semigroups (or band complexes, or foliated 2-complexes, or partial
isometries of multi-intervals). This process was independently
invented by different people to study equations in groups, actions
on trees, some questions in algorithmic topology and dynamical
systems.
We call this process Elimination Process.
Makanin (1982): Initial version of EP for solving equations.
Makanin’s EP gives a decision algorithm to verify consistency of a
given system - decidability of the Diophantine problem over free
groups.
Makanin introduced the fundamental notions: generalized
equations,
elementary and entire transformations,
notion of
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Equations in groups, Elimination process
I will talk about an (infinite) rewriting process of a certain type
that transforms formal systems of equations in groups or
semigroups (or band complexes, or foliated 2-complexes, or partial
isometries of multi-intervals). This process was independently
invented by different people to study equations in groups, actions
on trees, some questions in algorithmic topology and dynamical
systems.
We call this process Elimination Process.
Makanin (1982): Initial version of EP for solving equations.
Makanin’s EP gives a decision algorithm to verify consistency of a
given system - decidability of the Diophantine problem over free
groups.
Makanin introduced the fundamental notions: generalized
equations,
elementary and entire transformations,
notion of
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Equations in groups, Elimination process
I will talk about an (infinite) rewriting process of a certain type
that transforms formal systems of equations in groups or
semigroups (or band complexes, or foliated 2-complexes, or partial
isometries of multi-intervals). This process was independently
invented by different people to study equations in groups, actions
on trees, some questions in algorithmic topology and dynamical
systems.
We call this process Elimination Process.
Makanin (1982): Initial version of EP for solving equations.
Makanin’s EP gives a decision algorithm to verify consistency of a
given system - decidability of the Diophantine problem over free
groups.
Makanin introduced the fundamental notions: generalized
equations,
elementary and entire transformations,
notion of
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Razborov’s process
Razborov (1987): further developed EP to describe
algorithmically all solutions of a given system in F .
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Kharlampovich - Myasnikov (1998):
Refined Razborov’s process.
Effective (algorithmic) description of solutions of equations in free
groups in terms of triangular quadratic systems of equations.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Kharlampovich - Myasnikov (1998):
Refined Razborov’s process.
Effective (algorithmic) description of solutions of equations in free
groups in terms of triangular quadratic systems of equations.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
From the group theoretic view-point the elimination process tells
something about the coordinate groups of the systems involved.
This allows one to transform the pure combinatorial and
algorithmic results obtained in the elimination process into
statements about the coordinate groups.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
From equations to intervals
xyz=1
x
λ2
y
λ1
λ3
z
y
x
λ1
λ2
λ2
z
λ3
λ3
λ1
Figure: From the cancellation tree for the equation xyz = 1 to the
−1
−1
generalized equation (x = λ1 ◦ λ2 , y = λ−1
2 ◦ λ3 , z = λ3 ◦ λ1 ).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
A generalized equation can be seen as a collection of isometries
between subintervals of an unknown interval [0, N] ⊆ Z.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
[x,y][b,a]=1
y
x
a b
y
a
b
1
x
2
3
μ1
λ
λ1
4
b a
5
6
μ
λ2
7
8
μ
λ
λ1
9 10 11
μ1
a b
λ2
Figure: The cancellation tree and generalized equation for the equation
[x, y ][b, a] = 1, x = λ1 = λ1 , y = µ = µ1 ab = λ2 .
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Entire transformation
1
2
3
4
5
6
7
8
9
10
µ
µ
λ
λ
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Elimination process and JSJ
A splitting of G is a representation of G as the fundamental
groups of a graph of groups.
A splitting is cyclic (abelian) if all the edge groups are cyclic
(abelian).
Elementary splittings:
G = A ∗C B, G = A∗C = hA, t | t −1 Ct = C 0 i,
Free splittings:
G =A∗B
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Grushko’s decompositions
All free splittings of G are encoded in Grushko’s decompositions.
A free decomposition
G = G1 ∗ . . . ∗ Gk ∗ Fr
is a Grushko’s decomposition of G if G1 , . . . , Gk are freely
indecomposable non-cyclic groups and Fr is a free group of rank r .
Grushko’s decompositions are essentially unique.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
JSJ decompositions
All cyclic (abelian) splittings of G are encoded in JSJ
decompositions of G .
JSJ decompositions are universal decompositions with vertices of
the following types: QH-vertices, abelian, rigid.
JSJ decompositions are essentially unique.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Infinite branches and JSJ
Motto: JSJ is an algebraic counterpart of EP.
infinite branches of EP ⇐⇒ abelian splittings of the coordinate
groups of the systems.
Moreover, the automorphisms associated with infinite branches of
the process are precisely the canonical automorphisms of the JSJ
decomposition associated with the splittings.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Elimination Processes allow one to recognize and construct
effectively abelian splittings of the coordinate groups.
This is a tool to construct universal decompositions (JSJ
decompositions) of f.g. fully residually free groups (Khar,
Myasnikov) groups of automorphisms (Bumagin, Kh., Myasnikov),
to solve the isomorphism problem (Bumagin, Kh., Myasnikov).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Elimination Processes allow one to recognize and construct
effectively abelian splittings of the coordinate groups.
This is a tool to construct universal decompositions (JSJ
decompositions) of f.g. fully residually free groups (Khar,
Myasnikov) groups of automorphisms (Bumagin, Kh., Myasnikov),
to solve the isomorphism problem (Bumagin, Kh., Myasnikov).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Elimination Processes allow one to recognize and construct
effectively abelian splittings of the coordinate groups.
This is a tool to construct universal decompositions (JSJ
decompositions) of f.g. fully residually free groups (Khar,
Myasnikov) groups of automorphisms (Bumagin, Kh., Myasnikov),
to solve the isomorphism problem (Bumagin, Kh., Myasnikov).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Elimination Processes allow one to recognize and construct
effectively abelian splittings of the coordinate groups.
This is a tool to construct universal decompositions (JSJ
decompositions) of f.g. fully residually free groups (Khar,
Myasnikov) groups of automorphisms (Bumagin, Kh., Myasnikov),
to solve the isomorphism problem (Bumagin, Kh., Myasnikov).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Isomorphism problem
Theorem [Bumagin, Kharlampovich, M.]
The isomorphism problem is decidable in the class of all finitely
generated fully residually free groups.
Dahmani and Groves generalized this result to torsion-free
groups which are hyperbolic relative to abelian subgroups.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Isomorphism problem
Theorem [Bumagin, Kharlampovich, M.]
The isomorphism problem is decidable in the class of all finitely
generated fully residually free groups.
Dahmani and Groves generalized this result to torsion-free
groups which are hyperbolic relative to abelian subgroups.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Algorithmic topology
The first algorithm for UNKNOTTING was given by Haken (61), it
was shown by Haas, Lagarias, Pippenger (91) that UNKNOTING
is NP.
Problem 3-MANIFOLD KNOT GENUS (the minimal genus of an
orientable spanning surface for a knot in 3-dim manifold)
INSTANCE: A triangulated 3-dimensional manifold M, a knot K in
the 1-skeleton of M, and a natural number g .
QUESTION: Does the knot K have g (K ) ≤ g ?
Theorem (Agol, Hass, Thurston, 2002). 3-MANIFOLD KNOT
GENUS is NP complete.
The size of an instance is given by the sum of the number of
tetrahedra t in M and log g .
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
3-Manifold Knot Genus is NP Complete
A normal surface S in a triangulated compact 3-manifold M is a
PL-surface whose intersection with each tetrahedron in M consists
of a disjoint set of elementary disks. These are either triangles or
quadrilaterals.
Within each tetrahedron of M there are 4 possible triangles and 3
possible quadrilaterals. Haken observed that a normal surface is
determined by the number of pieces of each of the 7 kinds of
elementary disks that occur in each tetrahedron, or a non-negative
Olga Kharlampovich
(McGill University)
7t
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
3-Manifold Knot Genus is NP Complete
The Haken normal cone is specified by linear equations and
inequalities of the form
vi1 + vi2 = vi3 + vi4 (up to 6t equations),
vi ≥ 0 (1 ≤ i ≤ 7t).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
3-Manifold Knot Genus is NP Complete
NP-hardness is proved by reducing an instance of ONE-IN-THREE
SAT to an instance in 3-MANIFOLD KNOT GENUS.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
3-Manifold Knot Genus is NP Complete
The proof that 3-MANIFOLD KNOT GENUS is in NP is based on
Combinatorial procedure that computes the number of orbits
of a collection of k isometries between subintervals of an
interval [1, N] ⊂ Z in time polynomial in k(logN).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
3-Manifold Knot Genus is NP Complete
By lining up the intersections of a normal surface with edges of a
triangulation, they obtain such subintervals.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
3-Manifold Knot Genus is NP Complete
Then apply the orbit counting algorithm to count the number of
components of a normal surface F (=the number of components
of its boundary). Here N is the total number of points at which F
meets the 1-skeleton of MK .
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
NP completeness of the Diophantine problem for free
groups
Lysenok announces that the problem whether a finite system of
equations has a solution in a free group is NP complete.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on trees
Theorem. A group G is free if and only if it acts freely on a tree.
Free action = no inversion of edges and stabilizers of vertices are
trivial.
An R-tree is a metric space (X , p) (where p : X × X → R) which
satisfies the following properties:
1) (X , p) is geodesic,
2) if two segments of (X , p) intersect in a single point, which is
an endpoint of both, then their union is a segment,
3) the intersection of two segments with a common endpoint is
also a segment.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on trees
R-trees were introduced by J.Tits in 1977.
The theory of R-trees was developed by Morgan and Shalen
(1985), Culler and Morgan (1987).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on trees
Examples
X = R with usual metric.
A geometric realization of a simplicial tree.
X = R2 with metric d defined by
|y1 | + |y2 | + |x1 − x2 | if x1 6= x2
d((x1 , y1 ), (x2 , y2 )) =
|y1 − y2 |
if x1 = x2
French railway system.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on trees
Rips’ Theorem [Rips, 1991 - not published] A f.g. group is R-free if
and only if it is a free product of surface groups (except for the
non-orientable surfaces of genus 1,2, 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’ Theorem
proving a more general result for stable actions on R-trees.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on trees
Let G act freely on T . Let {γ1 , . . . , γp } be a system of generators
of G . K ⊂ T be a finite subtree. γi gives
gi : K ∩ γi−1 K → γi K ∩ K . If K large enough, then G may be read
off the dynamical system T = (K , {gi }): generators are γ1 , . . . , γp ,
relations are all words γi±1
. . . γi±1
such that there exists x ∈ K
n
1
±1
with gi±1
.
.
.
g
(x)
=
x
(staying
in
K ).
in
1
A simple manipulation allows us to replace a real tree T by a
real object of study: a system X consisting of finite disjoint
union D of compact subintervals of R, together with a finite
number of partially defined isometries φj : Aj → Bj , where
each base Aj , Bj is a compact subinterval of D. One may
associate a group G (X ) to a system X . Generators are φi ’s.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on trees
One may associate to X a free action of G (X ) on some R-tree T
provided X satisfies (a property similar to) the following “no
reflection condition”: there is no x such that x + t is in the orbit
of x − t for t > 0 small.
There is a compact foliated 2-complex Σ(X ), and G (X ) is
obtained from π1 Σ(X ) by killing all loops contained in leaves. Let
Σ0 (X ) → Σ(X ) be the covering with transformation group G (X ).
The tree T is the space of leaves of the lifted foliation on Σ0 (X ).
The absence of reflection implies that every leaf in Σ0 (X ) is closed.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Systems of isometries
If X is a system of isometries, we define G (X ) = π1 (Σ)/L̄, where
L̄ is the normal subgroups generated by the free homotopy classes
of loops contained in leaves.
Theorem (Rips) Let X be a connected system of isometries. Then
G (X ) is a free product of free abelian groups and surface groups.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
The Fundamental Problem
The following is a principal step in the Alperin-Bass’ program:
Open Problem [Rips, Bass]
Describe finitely generated groups acting freely on Λ-trees.
Here ”describe” means ”describe in the standard group-theoretic
terms”.
Λ-free groups = groups acting freely on Λ-trees.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
The Fundamental Problem
The following is a principal step in the Alperin-Bass’ program:
Open Problem [Rips, Bass]
Describe finitely generated groups acting freely on Λ-trees.
Here ”describe” means ”describe in the standard group-theoretic
terms”.
Λ-free groups = groups acting freely on Λ-trees.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
The Fundamental Problem
The following is a principal step in the Alperin-Bass’ program:
Open Problem [Rips, Bass]
Describe finitely generated groups acting freely on Λ-trees.
Here ”describe” means ”describe in the standard group-theoretic
terms”.
Λ-free groups = groups acting freely on Λ-trees.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Chiswell’s Theorem
Let (X , d) be a Λ-tree and x ∈ X . If G acts on (X , d) then
L(g ) = d(x, gx) is a Lyndon length function on G .
Theorem [Chiswell]
Let L : G → Λ be a Lyndon length function on a group G . Then
there exists a Λ-tree (X , d), x ∈ X , and an isometric action of G
on 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 (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Chiswell’s Theorem
Let (X , d) be a Λ-tree and x ∈ X . If G acts on (X , d) then
L(g ) = d(x, gx) is a Lyndon length function on G .
Theorem [Chiswell]
Let L : G → Λ be a Lyndon length function on a group G . Then
there exists a Λ-tree (X , d), x ∈ X , and an isometric action of G
on 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 (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Chiswell’s Theorem
Let (X , d) be a Λ-tree and x ∈ X . If G acts on (X , d) then
L(g ) = d(x, gx) is a Lyndon length function on G .
Theorem [Chiswell]
Let L : G → Λ be a Lyndon length function on a group G . Then
there exists a Λ-tree (X , d), x ∈ X , and an isometric action of G
on 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 (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Chiswell’s Theorem
Let (X , d) be a Λ-tree and x ∈ X . If G acts on (X , d) then
L(g ) = d(x, gx) is a Lyndon length function on G .
Theorem [Chiswell]
Let L : G → Λ be a Lyndon length function on a group G . Then
there exists a Λ-tree (X , d), x ∈ X , and an isometric action of G
on 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 (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Non-Archimedean actions
Theorem (H.Bass, 1991)
A finitely generated (Λ ⊕ Z)-free group is the fundamental group
of 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 -free
groups.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Non-Archimedean actions
Theorem (H.Bass, 1991)
A finitely generated (Λ ⊕ Z)-free group is the fundamental group
of 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 -free
groups.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on Rn -trees
Theorem [Guirardel, 2003]
A f.g. freely indecomposable Rn -free group is isomorphic to the
fundamental group of a finite graph of groups, where each vertex
group 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 abelian
subgroups.
Notice, that Zn -free groups are Rn -free.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on Rn -trees
Theorem [Guirardel, 2003]
A f.g. freely indecomposable Rn -free group is isomorphic to the
fundamental group of a finite graph of groups, where each vertex
group 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 abelian
subgroups.
Notice, that Zn -free groups are Rn -free.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on Rn -trees
Theorem [Guirardel, 2003]
A f.g. freely indecomposable Rn -free group is isomorphic to the
fundamental group of a finite graph of groups, where each vertex
group 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 abelian
subgroups.
Notice, that Zn -free groups are Rn -free.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Actions on Rn -trees
Theorem [Guirardel, 2003]
A f.g. freely indecomposable Rn -free group is isomorphic to the
fundamental group of a finite graph of groups, where each vertex
group 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 abelian
subgroups.
Notice, that Zn -free groups are Rn -free.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Zn -free groups
Theorem [Kharlampovich, Miasnikov, 96]
Finitely generated fully residually free groups are Zn -free.
Theorem [Martino and Rourke, 2005]
Let G1 and G2 be Zn -free groups. Then the amalgamated product
G1 ∗C G2 is Zm -free for some m ∈ N, provided C is cyclic and
maximal abelian in both factors.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Zn -free groups
Theorem [Kharlampovich, Miasnikov, 96]
Finitely generated fully residually free groups are Zn -free.
Theorem [Martino and Rourke, 2005]
Let G1 and G2 be Zn -free groups. Then the amalgamated product
G1 ∗C G2 is Zm -free for some m ∈ N, provided C is cyclic and
maximal abelian in both factors.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Examples of Zn -free groups:
R-free groups,
h x1 , x2 , x3 | x12 x22 x32 = 1 i is Z2 -free (but is neither R-free, nor
fully residually free).
The latter answers Conjecture Q 3.4 on Bestwina’s problem page
in the negative: it is not true that any Rn -free (or even Zn -free)
group is fully residually free.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Examples of Zn -free groups:
R-free groups,
h x1 , x2 , x3 | x12 x22 x32 = 1 i is Z2 -free (but is neither R-free, nor
fully residually free).
The latter answers Conjecture Q 3.4 on Bestwina’s problem page
in the negative: it is not true that any Rn -free (or even Zn -free)
group is fully residually free.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Examples of Zn -free groups:
R-free groups,
h x1 , x2 , x3 | x12 x22 x32 = 1 i is Z2 -free (but is neither R-free, nor
fully residually free).
The latter answers Conjecture Q 3.4 on Bestwina’s problem page
in the negative: it is not true that any Rn -free (or even Zn -free)
group is fully residually free.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
The Main Conjecture.
Conjecture
Every finitely generated Λ-free group is Zn -free.
All known finitely generated Λ-free groups are Zm -free.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
The Main Conjecture.
Conjecture
Every finitely generated Λ-free group is Zn -free.
All known finitely generated Λ-free groups are Zm -free.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Finitely presented complete Λ-free groups.
Theorem [Kharlampovich, Myasnikov, Serbin]
If G is f.p. and acts freely and regularly (all branch points are in
the same orbit) on a Λ-tree, then G has an index two subgroup
that can be represented as a 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 in
which associated subgroups are maximal abelian and
length-isomorphic.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Theorem [Kharlampovich, Myasnikov, Serbin]
Any f.p. group G that acts freely and regularly on a Λ-tree has an
index two subgroup with free length function in Zk ordered
lexicographically for an appropriate k ∈ N.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Interval exchange transformations
An interval exchange transformation is a kind of dynamical system
that generalises the idea of a circle rotation. The phase space
consists of the unit interval, and the transformation acts by cutting
the interval into several subintervals, and then permuting these
subintervals.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Interval exchange transformations
Let n > 0 and let π be a permutation on 1, . . . , n. Consider a
vector λ = (λ1 , . . . , λn ) of positive real numbers such that
λ1 + . . . + λn = 1. Define a map
Tπ,λ : [0, 1] → [0, 1]
called the interval exchange transformation associated to the pair
(π, λ) as follows. Let ai = Σ1≤j<i λj . Then
Thus Tπ,λ acts on each subinterval of the form [ai , ai + λi ) by an
orientation-preserving isometry, and it rearranges these subintervals
so that the subinterval at position i is moved to position π(i).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Interval exchange transformations
If n > 2, and provided π satisfies certain non-degeneracy
conditions, a deep theorem due independently to W.Veech and to
H.Masur asserts that for almost all choices of λ in the unit simplex
{λ = (λ1 , . . . , λn )|λ1 + . . . + λn = 1}
the transformation Tπ,λ uniquely ergodic (the orbits of points of
[0, 1] are uniformly evenly distributed).
However, for n ≥ 4 there also exist choices of (π, λ) so that it is
ergodic but not uniquely ergodic. Even in these cases, the number
of ergodic invariant measures of Tπ,λ is finite, and is at most n.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Interval exchange transformations)
The proof uses the so-called Rauzy-Veech induction= Entire
transformation for quadratic equations.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Linear Involutions with flips (Dantony, Nogueira, 1990
Let I be the interval I+ = I × {+1}, I− = I × {−1}. Suppose for
each base Ai ∈ I+ the dual Āi is on I− . Denote by T the interval
exchange transformation that acts on the interior of each Ai as φi
(φ◦i ).
Let λi be the length of the base µi .The system of isometries that
we consider is characterized by a permutation τ , a flip set F , and
λ = (λ1 , . . . , λn ). Let α = (τ, F ). We denote by Λα the set of
interval exchanges of type α. We can suppose that
λ1 + . . . + λn = 1, λi ≥ 0, i = 1, . . . , n (the standard n − 1
simplex).
The measure of Λα is 1. This is the volume of the pyramid in Rn
with the standard (n − 1)-simplex as a base and the origin as a
vertex.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Linear Involutions with flips (Dantony, Nogueira, 1990)
X is an interval exchange if X is connected, nondegenerate, and
every x ∈ D belongs to exactly two bases.
Theorem Almost all (with respect to some Lebesque measure)
interval exchanges with flip do not define a free action of a surface
group on a real tree (have a reflection).
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d
Equations in groups Elimination Processes
Algorithmic topology
Actions on trees
Finitely generated Λ-free groups
Dynamical systems
Consider the moduli space of pairs (C , ω), where C is a smooth
compact complex curve of a given genus and ω is a holomorphic
1-form on C with a given list of multiplicities of zeroes. Kontsevich
and Zorich describe connected components of this space. This
classification is important in the study of dynamics of interval
exchange transformations and billiards in rational polygons, and in
the study of geometry of translation surfaces.
Olga Kharlampovich (McGill University)
Equations in groups, actions on trees, algorithmic topology and d