The Grigorchuk and Grigorchuk-Machi Groups of Intermediate Growth - PowerPoint PPT Presentation

The Grigorchuk and Grigorchuk-Machi Groups of Intermediate Growth Levi Sledd Vanderbilt University July 20, 2019

References: 1. de la Harpe, Topics in Geometric Group Theory , Chapter VIII. 2. Grigorchuk, Machi, “A group of intermediate growth acting by homomorphisms on the real line.” If you want these slides you can email me: levi.sledd@vanderbilt.edu

The Free Monoid Definition Let A be a set. Then the free monoid on A , denoted A ∗ , is the set of all words over the alphabet A . More formally, the set of all finite sequences of elements of A .

The Free Monoid Definition Let A be a set. Then the free monoid on A , denoted A ∗ , is the set of all words over the alphabet A . More formally, the set of all finite sequences of elements of A . For w ∈ A ∗ , the length of w is denoted | w | . The empty word ε is defined to be the unique word of length 0.

The Free Monoid Definition Let A be a set. Then the free monoid on A , denoted A ∗ , is the set of all words over the alphabet A . More formally, the set of all finite sequences of elements of A . For w ∈ A ∗ , the length of w is denoted | w | . The empty word ε is defined to be the unique word of length 0. If G is a group generated by a finite set S , then we can evaluate words in ( S ∪ S − 1 ) ∗ to elements of G . Notation: w = G g .

The Free Monoid Definition Let A be a set. Then the free monoid on A , denoted A ∗ , is the set of all words over the alphabet A . More formally, the set of all finite sequences of elements of A . For w ∈ A ∗ , the length of w is denoted | w | . The empty word ε is defined to be the unique word of length 0. If G is a group generated by a finite set S , then we can evaluate words in ( S ∪ S − 1 ) ∗ to elements of G . Notation: w = G g . Definition The word length of an element g ∈ G with respect to S is | g | S = min {| w | | w = G g } .

Growth of Groups Definition Let G be a group generated by a finite set S . The growth of G with respect to S is the function γ G , S : N → N given by γ G , S ( n ) = |{ g ∈ G | | g | S ≤ n }| .

Growth of Groups Definition Let G be a group generated by a finite set S . The growth of G with respect to S is the function γ G , S : N → N given by γ G , S ( n ) = |{ g ∈ G | | g | S ≤ n }| . Recall that up to the asymptotic equivalence discussed in Supun’s talk, γ G , S is quasi-isometry invariant. In particular,

Growth of Groups Definition Let G be a group generated by a finite set S . The growth of G with respect to S is the function γ G , S : N → N given by γ G , S ( n ) = |{ g ∈ G | | g | S ≤ n }| . Recall that up to the asymptotic equivalence discussed in Supun’s talk, γ G , S is quasi-isometry invariant. In particular, ◮ γ G , S is independent of the choice of finite generating set S . Therefore we drop S subscripts from now on.

Growth of Groups Definition Let G be a group generated by a finite set S . The growth of G with respect to S is the function γ G , S : N → N given by γ G , S ( n ) = |{ g ∈ G | | g | S ≤ n }| . Recall that up to the asymptotic equivalence discussed in Supun’s talk, γ G , S is quasi-isometry invariant. In particular, ◮ γ G , S is independent of the choice of finite generating set S . Therefore we drop S subscripts from now on. ◮ If G is commensurate to H ( G ∼ H ), then γ G ∼ γ H .

Growth of Groups Definition Let G be a group generated by a finite set S . The growth of G with respect to S is the function γ G , S : N → N given by γ G , S ( n ) = |{ g ∈ G | | g | S ≤ n }| . Recall that up to the asymptotic equivalence discussed in Supun’s talk, γ G , S is quasi-isometry invariant. In particular, ◮ γ G , S is independent of the choice of finite generating set S . Therefore we drop S subscripts from now on. ◮ If G is commensurate to H ( G ∼ H ), then γ G ∼ γ H . Exercise: γ G × H ∼ γ G γ H .

Growth of Groups Definition Let G be a group generated by a finite set S . The growth of G with respect to S is the function γ G , S : N → N given by γ G , S ( n ) = |{ g ∈ G | | g | S ≤ n }| . Recall that up to the asymptotic equivalence discussed in Supun’s talk, γ G , S is quasi-isometry invariant. In particular, ◮ γ G , S is independent of the choice of finite generating set S . Therefore we drop S subscripts from now on. ◮ If G is commensurate to H ( G ∼ H ), then γ G ∼ γ H . Exercise: γ G × H ∼ γ G γ H . Lemma (VIII.61,63) G , then there exists an α ∈ (0 , 1) such that e n α � γ G . If γ G ∼ γ 2

The Infinite Rooted Binary Tree Let T be the infinite rooted binary tree.

The Infinite Rooted Binary Tree Let T be the infinite rooted binary tree. V ( T ) = { 0 , 1 } ∗ and u ∼ w if u = w ′ or w = u ′ .

The Infinite Rooted Binary Tree Let T be the infinite rooted binary tree. V ( T ) = { 0 , 1 } ∗ and u ∼ w if u = w ′ or w = u ′ . ε 0 1 00 01 10 11 000 001 010 011 100 101 110 111

Automorphisms of T ε Let α ∈ Aut( T ). 0 1 00 01 10 11 000 001 010 011 100 101 110 111

Automorphisms of T ε Let α ∈ Aut( T ). Since α preserves degrees of vertices, 0 1 α fixes the root ε . 00 01 10 11 000 001 010 011 100 101 110 111

Automorphisms of T ε Let α ∈ Aut( T ). Since α preserves degrees of vertices, 0 1 α fixes the root ε . Since α preserves distances (to ε ), α ( L n ) = L n and α ( T n ) = T n . 00 01 10 11 000 001 010 011 100 101 110 111

Automorphisms of T ε Let α ∈ Aut( T ). Since α preserves degrees of vertices, 0 1 α fixes the root ε . Since α preserves distances (to ε ), α ( L n ) = L n and α ( T n ) = T n . 00 01 10 11 Level by level, α independently switches ( s ) or fixes ( f ) the children of each vertex w in the level. 000 001 010 011 100 101 110 111

Automorphisms of T α s Let α ∈ Aut( T ). Since α preserves degrees of vertices, s f α fixes the root ε . Since α preserves distances (to ε ), α ( L n ) = L n and α ( T n ). s f s s Level by level, α independently switches ( s ) or fixes ( f ) the children of each vertex w in the level.

Automorphisms of T α s Level by level, α independently switches ( s ) or fixes ( f ) the children of each vertex w in the level. s f s f s s

Automorphisms of T α s Level by level, α independently switches ( s ) or fixes ( f ) the children of each vertex w in the level. In particular, | Aut( T ) | = 2 ℵ 0 . s f s f s s

Automorphisms of T α s Level by level, α independently switches ( s ) or fixes ( f ) the children of each vertex w in the level. In particular, | Aut( T ) | = 2 ℵ 0 . s f Let St n be the pointwise stabilizer of T n . Note: ◮ Aut( T ) / St n ∼ = Aut( T n ), so s f s s [St n : Aut( T )] < ∞ .

Automorphisms of T α s Level by level, α independently switches ( s ) or fixes ( f ) the children of each vertex w in the level. In particular, | Aut( T ) | = 2 ℵ 0 . s f Let St n be the pointwise stabilizer of T n . Note: ◮ Aut( T ) / St n ∼ = Aut( T n ), so s f s s [St n : Aut( T )] < ∞ . ◮ � ∞ n =1 St n = { 1 } .

Automorphisms of T α s Level by level, α independently switches ( s ) or fixes ( f ) the children of each vertex w in the level. In particular, | Aut( T ) | = 2 ℵ 0 . s f Let St n be the pointwise stabilizer of T n . Note: ◮ Aut( T ) / St n ∼ = Aut( T n ), so s f s s [St n : Aut( T )] < ∞ . ◮ � ∞ n =1 St n = { 1 } . Therefore Aut( T ) is residually finite.

Self-similarity of Aut(T) The action of α on the 0 and 1 subtrees produce elements α 0 , α 1 ∈ Aut( T ). α s α 0 α 1 s f s f s s

Self-similarity of Aut(T) The action of α on the 0 and 1 subtrees produce elements α 0 , α 1 ∈ Aut( T ). α s This gives us a homomorphism ψ 1 : α �→ ( α 0 , α 1 ). α 0 α 1 s f s f s s

Self-similarity of Aut(T) The action of α on the 0 and 1 subtrees produce elements α 0 , α 1 ∈ Aut( T ). α s This gives us a homomorphism ψ 1 : α �→ ( α 0 , α 1 ). Similarly, α 0 α 1 s f ψ n : Aut( T ) ։ Aut( T ) 2 n ψ n : α �→ ( α 0 ... 0 , . . . , α 1 ... 1 ) s f s s

Self-similarity of Aut(T) The action of α on the 0 and 1 subtrees produce elements α 0 , α 1 ∈ Aut( T ). α s This gives us a homomorphism ψ 1 : α �→ ( α 0 , α 1 ). Similarly, α 0 α 1 s f ψ n : Aut( T ) ։ Aut( T ) 2 n ψ n : α �→ ( α 0 ... 0 , . . . , α 1 ... 1 ) We have ψ 1 : Aut( T ) ։ Aut( T ) 2 and s f s s Ker( ψ 1 ) ∼ = Aut( T 1 ).

Self-similarity of Aut(T) The action of α on the 0 and 1 subtrees produce elements α 0 , α 1 ∈ Aut( T ). α s This gives us a homomorphism ψ 1 : α �→ ( α 0 , α 1 ). Similarly, α 0 α 1 s f ψ n : Aut( T ) ։ Aut( T ) 2 n ψ n : α �→ ( α 0 ... 0 , . . . , α 1 ... 1 ) We have ψ 1 : Aut( T ) ։ Aut( T ) 2 and s f s s Ker( ψ 1 ) ∼ = Aut( T 1 ). ψ 1 | St 1 : St 1 → Aut( T ) 2 is an isomorphism.

Self-similarity of Aut(T) The action of α on the 0 and 1 subtrees produce elements α 0 , α 1 ∈ Aut( T ). α s This gives us a homomorphism ψ 1 : α �→ ( α 0 , α 1 ). Similarly, α 0 α 1 s f ψ n : Aut( T ) ։ Aut( T ) 2 n ψ n : α �→ ( α 0 ... 0 , . . . , α 1 ... 1 ) We have ψ 1 : Aut( T ) ։ Aut( T ) 2 and s f s s Ker( ψ 1 ) ∼ = Aut( T 1 ). ψ 1 | St 1 : St 1 → Aut( T ) 2 is an isomorphism. Therefore Aut( T ) ∼ Aut( T ) 2 .