Tiling approach to the study of iterated monodromy groups Mikhail - PowerPoint PPT Presentation

Tiling approach to the study of iterated monodromy groups Mikhail Hlushchanka (UCLA) joint with Daniel Meyer (University of Liverpool) University of Hawai’i at M¯ anoa March 23, 2019 M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 1 / 23

Outline 1 Main example and setup 2 Iterated monodromy groups 3 Growth of groups 4 Tiling approach to understanding of IMGs M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 2 / 23

Main example f T ≅ S 2 ≅ S 2 M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 3 / 23

Main example f T ≅ S 2 ≅ S 2 Constructed a (continuous) map f ∶ S 2 → S 2 . Note that f − 1 ( ∂ T ) ⊃ ∂ T . M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 3 / 23

Main example f T ≅ S 2 ≅ S 2 Constructed a (continuous) map f ∶ S 2 → S 2 . Note that f − 1 ( ∂ T ) ⊃ ∂ T . Used by D. Meyer to study a triangular “snowball”, a 3D analog of snowflake. M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 3 / 23

Setup The constructed map f ∶ S 2 → S 2 be a branched covering map, i.e., f is continuous; surjective; locally z ↦ z k , k ∈ N , after homeomorphic coordinate changes. f M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 4 / 23

Setup The constructed map f ∶ S 2 → S 2 be a branched covering map, i.e., f is continuous; surjective; locally z ↦ z k , k ∈ N , after homeomorphic coordinate changes. ○ f ○ ○ ○ ○ ○ ○ ○ ○ The critical set of f is crit ( f ) = { c ∈ S 2 ∶ deg ( f , c ) > 1 } . The postcritical set of f is post ( f ) = ⋃ ∞ n = 1 f n ( crit ( f )) . M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 4 / 23

Thurston maps Definition f ∶ S 2 → S 2 is called a Thurston map if f is an orientation-preserving branched cover of S 2 with d ∶ = deg ( f ) ≥ 2; postcritically finite (pcf), i.e., #post ( f ) < ∞ . M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 5 / 23

Thurston maps Definition f ∶ S 2 → S 2 is called a Thurston map if f is an orientation-preserving branched cover of S 2 with d ∶ = deg ( f ) ≥ 2; postcritically finite (pcf), i.e., #post ( f ) < ∞ . The constructed map f can “be realized by” (is combinatorially equivalent to) a rational map on the Riemann sphere ̂ C , that is, there exits rational map F ∶ ̂ C → ̂ C such that that f and F commute up to an isotopy relative to the postcritical set (Thurston’s characterization of rational maps). 8 ) 2 + 1 = 2 ( z 2 − 3 z 2 − 1 4 ) 3 3 F ( z ) = 2 ( 3 4 ) 8 ) 2 − 1 . z 2 ( z 2 − 9 z 2 ( z 2 − 9 M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 5 / 23

Outline 1 Main example and setup 2 Iterated monodromy groups 3 Growth of groups 4 Tiling approach to understanding of IMGs M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 6 / 23

Dynamical pre-image tree Let f ∶ S 2 → S 2 be a Thurston map of degree d (e.g., a pcf rational map). Let t ∈ S 2 ∖ post ( f ) . Consider the dynamical pre-image tree X 3 t ′ X 0 = { t } f ( t ′ ) X 2 X n = f − n ( t ) contains d n elements T = ⊔ X n is called the pre-image tree. X 1 n ≥ 0 t ′ ∈ X n and f ( t ′ ) ∈ X n − 1 are connected by an edge. X 0 t M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 7 / 23

Iterated monodromy action G ∶ = π 1 ( S 2 ∖ post ( f ) , t ) acts on the pre-image tree T by automorphisms via iterated monodromy action. M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 8 / 23

Iterated monodromy action G ∶ = π 1 ( S 2 ∖ post ( f ) , t ) acts on the pre-image tree T by automorphisms via iterated monodromy action. Consider a loop γ ⊂ S 2 ∖ post ( f ) at t and a point t ′ ∈ X n = f − n ( t ) . ̃ ( t ′ ) γ t ′ γ γ t M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 8 / 23

Iterated monodromy action G ∶ = π 1 ( S 2 ∖ post ( f ) , t ) acts on the pre-image tree T by automorphisms via iterated monodromy action. Consider a loop γ ⊂ S 2 ∖ post ( f ) at t and a point t ′ ∈ X n = f − n ( t ) . Since f n ∶ S 2 ∖ f − n ( post ( f )) → S 2 ∖ post ( f ) is ̃ ( t ′ ) γ t ′ γ a covering map, can lift γ by f n starting at t ′ to a curve ̃ γ . γ t M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 8 / 23

Iterated monodromy action G ∶ = π 1 ( S 2 ∖ post ( f ) , t ) acts on the pre-image tree T by automorphisms via iterated monodromy action. Consider a loop γ ⊂ S 2 ∖ post ( f ) at t and a point t ′ ∈ X n = f − n ( t ) . Since f n ∶ S 2 ∖ f − n ( post ( f )) → S 2 ∖ post ( f ) is ̃ ( t ′ ) γ t ′ γ a covering map, can lift γ by f n starting at t ′ to a curve ̃ γ . Set ( t ′ ) γ ∶ = the endpoint of ̃ γ . γ t M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 8 / 23

Iterated monodromy action G ∶ = π 1 ( S 2 ∖ post ( f ) , t ) acts on the pre-image tree T by automorphisms via iterated monodromy action. Consider a loop γ ⊂ S 2 ∖ post ( f ) at t and a point t ′ ∈ X n = f − n ( t ) . Since f n ∶ S 2 ∖ f − n ( post ( f )) → S 2 ∖ post ( f ) is ̃ ( t ′ ) γ t ′ γ a covering map, can lift γ by f n starting at t ′ to a curve ̃ γ . Set ( t ′ ) γ ∶ = the endpoint of ̃ γ . Note: ( t ′ ) γ ∈ X n and it depends only on [ γ ] ∈ π 1 ( S 2 ∖ post ( f ) , t ) . γ t M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 8 / 23

Iterated monodromy group Formally, there is a group homomorphism ϕ ∶ G → Aut ( T ) M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 9 / 23

Iterated monodromy group Formally, there is a group homomorphism ϕ ∶ G → Aut ( T ) Definition (Nekrahevych’03) The iterated monodromy group of f is IMG ( f ) ∶ = G / ker ϕ ≃ ϕ ( G ) . M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 9 / 23

Iterated monodromy group Formally, there is a group homomorphism ϕ ∶ G → Aut ( T ) Definition (Nekrahevych’03) The iterated monodromy group of f is IMG ( f ) ∶ = G / ker ϕ ≃ ϕ ( G ) . IMG ( f ) is a self-similar group (w.r.t. some T v g T v natural labeling of T ): for each g ∈ IMG ( f ) and v ∈ T , v g v g ∣ T v ∶ T v → T v g corresponds to an element g g ∣ v ∈ IMG ( f ) . This allows to encode the action of g on T using finite combinatorial data. M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 9 / 23

Reasons to study IMG’s 1 a powerful invariant for pcf rational maps. For instance, IMG ( f ) recovers the Julia set J f and f ∶ J f → J f . M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 10 / 23

Reasons to study IMG’s 1 a powerful invariant for pcf rational maps. For instance, IMG ( f ) recovers the Julia set J f and f ∶ J f → J f . 2 a powerful algebraic tool to answer dynamical questions: ▸ Hubbard’s twisted rabbit problem; ▸ Global curve attractor problem. M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 10 / 23

Reasons to study IMG’s 1 a powerful invariant for pcf rational maps. For instance, IMG ( f ) recovers the Julia set J f and f ∶ J f → J f . 2 a powerful algebraic tool to answer dynamical questions: ▸ Hubbard’s twisted rabbit problem; ▸ Global curve attractor problem. 3 inspire new constructions and methods in group theory. ▸ limit spaces ( ≈ Julia sets) of contracting self-similar groups as a Gromov-Hausdorff limit of Schreier graphs on levels of T or the boundary of a Gromov hyperbolic space associated with these graphs. M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 10 / 23

Reasons to study IMG’s 1 a powerful invariant for pcf rational maps. For instance, IMG ( f ) recovers the Julia set J f and f ∶ J f → J f . 2 a powerful algebraic tool to answer dynamical questions: ▸ Hubbard’s twisted rabbit problem; ▸ Global curve attractor problem. 3 inspire new constructions and methods in group theory. ▸ limit spaces ( ≈ Julia sets) of contracting self-similar groups as a Gromov-Hausdorff limit of Schreier graphs on levels of T or the boundary of a Gromov hyperbolic space associated with these graphs. 4 provide groups with exotic algebraic properties: ▸ groups of intermediate growth, e.g., IMG ( z 2 + i ) ; ▸ amenable groups of exponential growth, e.g., IMG ( z 2 − 1 ) . M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 10 / 23

Outline 1 Main example and setup 2 Iterated monodromy groups 3 Growth of groups 4 Tiling approach to understanding of IMGs M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 11 / 23

Growth of groups Let G be a finitely generated group with a generating set S = { s 1 ,..., s k } . Let ℓ S ( g ) be the length of the element g ∈ G w.r.t. the set S , i.e., ℓ S ( g ) ∶= min { n ∈ N 0 ∶ g = s ε 1 n , where s j ∈ S and ε j ∈ { 1 , − 1 }} . 1 ... s ε n M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 12 / 23

Growth of groups Let G be a finitely generated group with a generating set S = { s 1 ,..., s k } . Let ℓ S ( g ) be the length of the element g ∈ G w.r.t. the set S , i.e., ℓ S ( g ) ∶= min { n ∈ N 0 ∶ g = s ε 1 n , where s j ∈ S and ε j ∈ { 1 , − 1 }} . 1 ... s ε n The growth function of G w.r.t. S is defined by gr G , S ( n ) = # { g ∈ G ∶ ℓ S ( g ) ≤ n } . M. Hlushchanka, D. Meyer Tilings and IMG’s March 23, 2019 12 / 23