Thompson’s group F Thompson’s group F is the group PL 2 ( I ), with respect to composition, of all piecewise-linear homeomorphisms of the unit interval I = [0 , 1] with a finite number of breakpoints, such that ◮ all slopes are integral powers of 2, ◮ all breakpoints have dyadic rational coordinates.
Thompson’s group T
Thompson’s group T Similar to F , but defined on the unit circle: it preserves the cyclic order of the intervals
Thompson’s group T Similar to F , but defined on the unit circle: it preserves the cyclic order of the intervals
Thompson’s group T Similar to F , but defined on the unit circle: it preserves the cyclic order of the intervals
Thompson’s group V
Thompson’s group V Similar to F , but not continuous: it permutes the order of the intervals and can be seen as a group of homeomorphisms of the Cantor set C to itself:
Thompson’s group V Similar to F , but not continuous: it permutes the order of the intervals and can be seen as a group of homeomorphisms of the Cantor set C to itself:
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
Multiplication of tree pairs
The group QAut ( T 2 , c )
The group QAut ( T 2 , c ) T 2 , c is the infinite binary 2-colored tree (left = red, right = blue).
The group QAut ( T 2 , c ) T 2 , c is the infinite binary 2-colored tree (left = red, right = blue). Definition QAut ( T 2 , c ) is the group of all maps T 2 , c → T 2 , c which respect the edge and color relation, except for possibly finitely many vertices.
The group QAut ( T 2 , c )
The group QAut ( T 2 , c ) ε ε 00 ε 0 0 1 1 0 00 00 01 01 10 10 11 11 000 01 01 1 001 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 0000 0001 0010 0011 010 010 011 011 10 11
Lehnert’s conjecture
Lehnert’s conjecture Theorem (Lehnert) QAut ( T 2 , c ) is in co CF .
Lehnert’s conjecture Theorem (Lehnert) QAut ( T 2 , c ) is in co CF . Conjecture (Lehnert) QAut ( T 2 , c ) is a universal co CF group.
The relation between V and QAut ( T 2 , c )
The relation between V and QAut ( T 2 , c ) Theorem (Lehnert) V ֒ → QAut ( T 2 , c ) .
The relation between V and QAut ( T 2 , c ) Theorem (Lehnert) V ֒ → QAut ( T 2 , c ) . Our version of his proposed embedding:
The relation between V and QAut ( T 2 , c ) Theorem (Lehnert) V ֒ → QAut ( T 2 , c ) . Our version of his proposed embedding: ◮ Given a tree T , regard it as a subtree of T 2 , c with root 0 (left child of the root of T 2 , c )
The relation between V and QAut ( T 2 , c ) Theorem (Lehnert) V ֒ → QAut ( T 2 , c ) . Our version of his proposed embedding: ◮ Given a tree T , regard it as a subtree of T 2 , c with root 0 (left child of the root of T 2 , c ) ◮ Define a bijection ω T : { leaves of T } → { nodes of T } ∪ { ε } in the left-to-right order so the the rightmost leaf goes to ε .
The relation between V and QAut ( T 2 , c ) Theorem (Lehnert) V ֒ → QAut ( T 2 , c ) . Our version of his proposed embedding: ◮ Given a tree T , regard it as a subtree of T 2 , c with root 0 (left child of the root of T 2 , c ) ◮ Define a bijection ω T : { leaves of T } → { nodes of T } ∪ { ε } in the left-to-right order so the the rightmost leaf goes to ε . ◮ Given ( D , R , σ ) ∈ V define its image this way: 1. σ takes subtrees of T 2 , c at leaves D to those at leaves of R . 2. If n is a node of D or the root of T 2 , c , send it to n ω − 1 D σω R .
The relation between V and QAut ( T 2 , c ) Theorem (Lehnert) V ֒ → QAut ( T 2 , c ) . Our version of his proposed embedding: ◮ Given a tree T , regard it as a subtree of T 2 , c with root 0 (left child of the root of T 2 , c ) ◮ Define a bijection ω T : { leaves of T } → { nodes of T } ∪ { ε } in the left-to-right order so the the rightmost leaf goes to ε . ◮ Given ( D , R , σ ) ∈ V define its image this way: 1. σ takes subtrees of T 2 , c at leaves D to those at leaves of R . 2. If n is a node of D or the root of T 2 , c , send it to n ω − 1 D σω R . Corollary (Lehnert-Schweitzer) Thompson’s group V is in co CF .
The relation between V and QAut ( T 2 , c )
The relation between V and QAut ( T 2 , c ) Lemma (Lehnert, Bleak-M-Neunh¨ offer) If τ ∈ QAut ( T 2 , c ) there is a pair d τ = ( v τ , p τ ) representing τ such that ◮ v τ ∈ V acts like τ beneath a suitable level (V -part ), ◮ p τ is a bijection on the nodes above ( bijection part ).
The relation between V and QAut ( T 2 , c ) Lemma (Lehnert, Bleak-M-Neunh¨ offer) If τ ∈ QAut ( T 2 , c ) there is a pair d τ = ( v τ , p τ ) representing τ such that ◮ v τ ∈ V acts like τ beneath a suitable level (V -part ), ◮ p τ is a bijection on the nodes above ( bijection part ). We call d τ a disjoint decomposition .
The relation between V and QAut ( T 2 , c ) Lemma (Lehnert, Bleak-M-Neunh¨ offer) If τ ∈ QAut ( T 2 , c ) there is a pair d τ = ( v τ , p τ ) representing τ such that ◮ v τ ∈ V acts like τ beneath a suitable level (V -part ), ◮ p τ is a bijection on the nodes above ( bijection part ). We call d τ a disjoint decomposition . There are many disjoint decompositions, but we can always define a minimal one (in some sense).
The relation between V and QAut ( T 2 , c )
The relation between V and QAut ( T 2 , c ) Question (Lehnert-Schweitzer) Does QAut ( T 2 , c ) embed into V ?
The relation between V and QAut ( T 2 , c ) Question (Lehnert-Schweitzer) Does QAut ( T 2 , c ) embed into V ? Theorem (Bleak-M-Neunh¨ offer) Yes.
The relation between V and QAut ( T 2 , c ) Question (Lehnert-Schweitzer) Does QAut ( T 2 , c ) embed into V ? Theorem (Bleak-M-Neunh¨ offer) Yes. Idea of the embedding: start with τ ∈ QAut ( T 2 , c ):
The relation between V and QAut ( T 2 , c ) Question (Lehnert-Schweitzer) Does QAut ( T 2 , c ) embed into V ? Theorem (Bleak-M-Neunh¨ offer) Yes. Idea of the embedding: start with τ ∈ QAut ( T 2 , c ): ◮ Build d τ = ( v τ , p τ ) with v τ = ( D τ , R τ , σ τ ),
The relation between V and QAut ( T 2 , c ) Question (Lehnert-Schweitzer) Does QAut ( T 2 , c ) embed into V ? Theorem (Bleak-M-Neunh¨ offer) Yes. Idea of the embedding: start with τ ∈ QAut ( T 2 , c ): ◮ Build d τ = ( v τ , p τ ) with v τ = ( D τ , R τ , σ τ ), ◮ Build a new tree pair ( � D τ , � R τ , � σ τ ) by “expanding v τ ” suitably.
The relation between V and QAut ( T 2 , c )
The relation between V and QAut ( T 2 , c ) ◮ Replace every node w in D d τ by a caret ( w , w n , w p ),
The relation between V and QAut ( T 2 , c ) ◮ Replace every node w in D d τ by a caret ( w , w n , w p ), Node with address w... ... becomes a caret in tree for V element. (But, not at address w!) w w w w n p
The relation between V and QAut ( T 2 , c ) ◮ Replace every node w in D d τ by a caret ( w , w n , w p ), Node with address w... ... becomes a caret in tree for V element. (But, not at address w!) w w w w n p ◮ If e parent , e left , e right are the edges attached to w , attach e left and e right to the bottom of w n and e parent to the top of w ,
The relation between V and QAut ( T 2 , c )
The relation between V and QAut ( T 2 , c ) ◮ Apply σ d τ to the n -leaves and b d τ to the p -leaves.
The relation between V and QAut ( T 2 , c ) ◮ Apply σ d τ to the n -leaves and b d τ to the p -leaves. ε ε 00 ε 0 0 1 1 0 00 00 01 01 10 10 11 11 000 01 01 1 001 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 0000 0001 0010 0011 010 010 011 011 10 11
The relation between V and QAut ( T 2 , c ) ◮ Apply σ d τ to the n -leaves and b d τ to the p -leaves. ε ε ε n ε n ε p ε p h b 0 1 0 1 0 0 1 1 0 0 1 1 n p n p n p n p f g e a h e 00 01 10 11 00 00 01 01 10 10 11 11 n p n p n p n p a b c d c d f g
The relation between V and QAut ( T 2 , c ) ◮ Apply σ d τ to the n -leaves and b d τ to the p -leaves. ε ε ε n ε n ε p ε p h b 0 1 0 1 0 0 1 1 0 0 1 1 n p n p n p n p f g e a h e 00 01 10 11 00 00 01 01 10 10 11 11 n p n p n p n p a b c d c d f g Lehnert’s conjecture revisited Thompson’s group V is the universal coCF group.
Work in progress on other subgroups of V
Work in progress on other subgroups of V We are working on embedding other subgroups into V .
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 : � a 1 , b 1 , . . . , a n , b n | [ a 1 , b 1 ] . . . [ a n , b n ] � (orientable) � a 1 , . . . , a n | a 2 1 . . . a 2 n � (non-orientable)
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 : � a 1 , b 1 , . . . , a n , b n | [ a 1 , b 1 ] . . . [ a n , b n ] � (orientable) � a 1 , . . . , a n | a 2 1 . . . a 2 n � (non-orientable) Recall: ◮ finite index subgroups of surface groups are still surface groups, ◮ there exist orientable double covers of non-orientable surfaces.
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