embeddings into thompson's group v and cocf groups · 2013. 8. 15. · zz2 does not embed into...

Embeddings into Thompson’s group V and coCFgroups

Francesco Matucci (joint with C. Bleak, M. Neunhoffer)

Groups St. Andrews 2013

St. Andrews

August 7, 2013

Finite state automaton

Mathematical model of computation to design computer programs.

Definition (DFSA)

A (deterministic) finite state automaton is a quintuple(S ,A, µ,Y , s0), where

I S is a finite set, called the state set,

I A is a finite set, called the alphabet,

I µ : A× S → S is a function, called the transition function,

I Y is a (possibly empty) subset of S called the subset ofaccept states,

I s0 ∈ S is called the start state.

If we allow µ : A× S → P(S), we call it a non-deterministic FSA.

RemarkEvery NDFSA is equivalent to a DFSA.

Definition (DFSA)

Finite state automaton: an example

Start0

1Accept

Finite state automaton: an example

Start0

1Accept

Regular language and finite groups

DefinitionGiven a FSA a regular language is the language given by all pathsinside the FSA which begin at the start state and end at an acceptstate.

DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe word problem is

WP(G ) = {words w in the monoid of X ∪ X−1 such that w ≡G 1}.

Theorem (Anisimov)

A finitely generated group G is finite if and only if WP(G ) is aregular language.

Theorem (Anisimov)

Pushdown automaton

Definition (PDA - Handwaving)

A PDA is like a FSA but it also employs a stack in its transitionfunction. The transition function pops and pushes a symbol at thetop of the stack and uses it to decide which state to reach.

RemarkPDA adds the stack as a parameter for choice. Finite statemachines just look at the input signal and the current state: theyhave no stack to work with.

It is not true that all NDPDA are equivalent to DPDA.

Pushdown automaton

Pushdown automaton: an example

finite state

automaton

Pushdown automaton: an example

finite state

automaton

Context free languages and virtually free groups

DefinitionContext-free languages are those accepted by PDAs.

DefinitionA finitely generated group is a context-free group if WP(G ) is acontext-free language.

Theorem (Muller-Schupp)

A finitely generated group G is virtually free if and only if it iscontext-free.

Co-context-free (Co-CF) groups

DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe coword problem is

coWP(G ) = {words w in the monoid of X ∪ X−1 such that w 6≡G 1}.

DefinitionA finitely generated group is a co-context-free group (coCF) ifcoWP(G ) is a context-free language. Let coCF be the class of allcoCF groups.

RemarkEvery CF group is in coCF (the converse is not true).

Closure properties of the class coCF

Theorem (Holt-Rover-Rees-Thomas)

The class coCF is closed under taking

I taking finite direct products,

I taking restricted standard wreath products with context-freetop groups,

I passing to finitely generated subgroups

I passing to finite index overgroups.

Conjecture (Holt-Rover-Rees-Thomas)coCF is not closed for free products. Candidate: Z ∗ Z2.

Theorem (Bleak-Salazar)

Z ∗ Z2 does not embed into Thompson’s group V .

Thompson’s group F

Thompson’s group F is the group PL2(I ), with respect tocomposition, of all piecewise-linear homeomorphisms of the unitinterval I = [0, 1] with a finite number of breakpoints, such that

I all slopes are integral powers of 2,

I all breakpoints have dyadic rational coordinates.

Thompson’s group T

Similar to F , but defined on the unit circle: it preserves the cyclicorder of the intervals

Thompson’s group V

Similar to F , but not continuous: it permutes the order of theintervals and can be seen as a group of homeomorphisms of theCantor set C to itself:

Thompson’s group V

Similar to F , but not continuous: it permutes the order of theintervals and can be seen as a group of homeomorphisms of theCantor set C to itself:

Multiplication of tree pairs

The group QAut(T2,c)

T2,c is the infinite binary 2-colored tree (left = red, right = blue).

DefinitionQAut(T2,c) is the group of all maps T2,c → T2,c which respect theedge and color relation, except for possibly finitely many vertices.

00 01 10 11

000 001 010 011 100 101 110 111

00 01 10 11

000 001 010 011 100 101 110 111

0000 0001 0010 0011 10 11

000 001 01

010 011

00 01 10 11

000 001 010 011 100 101 110 111

00 01 10 11

000 001 010 011 100 101 110 111

0000 0001 0010 0011 10 11

000 001 01

010 011

Lehnert’s conjecture

Theorem (Lehnert)

QAut(T2,c) is in coCF .

Conjecture (Lehnert)QAut(T2,c) is a universal coCF group.

Theorem (Lehnert)

The relation between V and QAut(T2,c)

Theorem (Lehnert)

V ↪→ QAut(T2,c).

Our version of his proposed embedding:

I Given a tree T , regard it as a subtree of T2,c with root 0 (leftchild of the root of T2,c)

I Define a bijection ωT : {leaves of T} → {nodes of T} ∪ {ε}in the left-to-right order so the the rightmost leaf goes to ε.

I Given (D,R, σ) ∈ V define its image this way:

1. σ takes subtrees of T2,c at leaves D to those at leaves of R.

2. If n is a node of D or the root of T2,c , send it to nω−1D σωR .

Corollary (Lehnert-Schweitzer)

Thompson’s group V is in coCF .

Theorem (Lehnert)

Lemma (Lehnert, Bleak-M-Neunhoffer)

If τ ∈ QAut(T2,c) there is a pair dτ = (vτ , pτ ) representing τ suchthat

I vτ ∈ V acts like τ beneath a suitable level (V -part),

I pτ is a bijection on the nodes above ( bijection part).

We call dτ a disjoint decomposition.

There are many disjoint decompositions, but we can always definea minimal one (in some sense).

Question (Lehnert-Schweitzer)Does QAut(T2,c) embed into V ?

Theorem (Bleak-M-Neunhoffer)

Idea of the embedding: start with τ ∈ QAut(T2,c):

I Build dτ = (vτ , pτ ) with vτ = (Dτ ,Rτ , στ ),

I Build a new tree pair (Dτ , Rτ , στ ) by “expanding vτ” suitably.

I Replace every node w in Ddτ by a caret (w ,wn,wp),

Node with address w... ... becomes a caret in tree for V element.

(But, not at address w!)

I If eparent, eleft, eright are the edges attached to w , attach eleftand eright to the bottom of wn and eparent to the top of w ,

I Apply σdτ to the n-leaves and bdτ to the p-leaves.

00 01 10 11

000 001 010 011 100 101 110 111

00 01 10 11

000 001 010 011 100 101 110 111

0000 0001 0010 0011 10 11

000 001 01

010 011

εn εp

0n 0p 1n 1p

00 p00 n01 p01

εn εp

0n 0p 1n 1p

10 p10 n11 p11

a b c d

εn εp

0n 0p 1n 1p

00 p00 n01 p01

εn εp

0n 0p 1n 1p

10 p10 n11 p11

a b c d

Lehnert’s conjecture revisitedThompson’s group V is the universal coCF group.

Work in progress on other subgroups of V

We are working on embedding other subgroups into V .

Candidates we are looking at are surface groups:

〈a1, b1, . . . , an, bn | [a1, b1] . . . [an, bn]〉 (orientable)

〈a1, . . . , an | a21 . . . a2n〉 (non-orientable)

Recall:

I finite index subgroups of surface groups are still surfacegroups,

I there exist orientable double covers of non-orientable surfaces.

Let H ≤ V . Any of its finite index overgroups is a subgroup of V .

Recall:

We are attempting to build a surface group inside V .

If one exists, then every other surface group will be in V .

QuestionDo surface groups embed in V ?

Surface groups are special cases of these Fuchsian groups:

〈a1, b1, . . . , an, bn, c1, . . . , ct , | cγ11 , . . . , cγtt ,

c−11 . . . c−1