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 Hawaii at M anoa March 23, 2019 M. Hlushchanka, D. Meyer Tilings and IMGs March 23,
Mikhail Hlushchanka (UCLA) joint with Daniel Meyer (University of Liverpool)
University of Hawai’i at M¯ anoa
March 23, 2019
Tilings and IMG’s March 23, 2019 1 / 23
1 Main example and setup 2 Iterated monodromy groups 3 Growth of groups 4 Tiling approach to understanding of IMGs
Tilings and IMG’s March 23, 2019 2 / 23
f ≅ S2 T ≅ S2
Tilings and IMG’s March 23, 2019 3 / 23
f ≅ S2 T ≅ S2 Constructed a (continuous) map f ∶ S2 → S2. Note that f −1(∂T) ⊃ ∂T.
Tilings and IMG’s March 23, 2019 3 / 23
f ≅ S2 T ≅ S2 Constructed a (continuous) map f ∶ S2 → S2. Note that f −1(∂T) ⊃ ∂T. Used by D. Meyer to study a triangular “snowball”, a 3D analog of snowflake.
Tilings and IMG’s March 23, 2019 3 / 23
The constructed map f ∶S2 → S2 be a branched covering map, i.e., f is continuous; surjective; locally z ↦ zk, k ∈ N, after homeomorphic coordinate changes. f
Tilings and IMG’s March 23, 2019 4 / 23
The constructed map f ∶S2 → S2 be a branched covering map, i.e., f is continuous; surjective; locally z ↦ zk, k ∈ N, after homeomorphic coordinate changes.
f The critical set of f is crit(f ) = {c ∈ S2 ∶ deg(f ,c) > 1}. The postcritical set of f is post(f ) = ⋃∞
n=1 f n(crit(f )).
Tilings and IMG’s March 23, 2019 4 / 23
Definition
f ∶S2 → S2 is called a Thurston map if f is an orientation-preserving branched cover of S2 with d ∶= deg(f ) ≥ 2; postcritically finite (pcf), i.e., #post(f ) < ∞.
Tilings and IMG’s March 23, 2019 5 / 23
Definition
f ∶S2 → S2 is called a Thurston map if f is an orientation-preserving branched cover of S2 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). F(z) = 2(3 4)
3
z2 − 1 z2(z2 − 9
8)2 + 1 = 2(z2 − 3 4)3
z2(z2 − 9
8)2 − 1.
Tilings and IMG’s March 23, 2019 5 / 23
1 Main example and setup 2 Iterated monodromy groups 3 Growth of groups 4 Tiling approach to understanding of IMGs
Tilings and IMG’s March 23, 2019 6 / 23
Let f ∶S2 → S2 be a Thurston map of degree d (e.g., a pcf rational map). Let t ∈ S2 ∖ post(f ). Consider the dynamical pre-image tree X 0 = {t} X n = f −n(t) contains dn elements T = ⊔
n≥0
X n is called the pre-image tree. t′ ∈ X n and f (t′) ∈ X n−1 are connected by an edge. t′ f (t′) X 3 X 2 X 1 X 0 t
Tilings and IMG’s March 23, 2019 7 / 23
G ∶= π1(S2 ∖ post(f ),t) acts on the pre-image tree T by automorphisms via iterated monodromy action.
Tilings and IMG’s March 23, 2019 8 / 23
G ∶= π1(S2 ∖ post(f ),t) acts on the pre-image tree T by automorphisms via iterated monodromy action. γ ̃ γ t′ (t′)γ t Consider a loop γ ⊂ S2 ∖ post(f ) at t and a point t′ ∈ X n = f −n(t).
Tilings and IMG’s March 23, 2019 8 / 23
G ∶= π1(S2 ∖ post(f ),t) acts on the pre-image tree T by automorphisms via iterated monodromy action. γ ̃ γ t′ (t′)γ t Consider a loop γ ⊂ S2 ∖ post(f ) at t and a point t′ ∈ X n = f −n(t). Since f n∶S2 ∖f −n(post(f )) → S2 ∖post(f ) is a covering map, can lift γ by f n starting at t′ to a curve ̃ γ.
Tilings and IMG’s March 23, 2019 8 / 23
G ∶= π1(S2 ∖ post(f ),t) acts on the pre-image tree T by automorphisms via iterated monodromy action. γ ̃ γ t′ (t′)γ t Consider a loop γ ⊂ S2 ∖ post(f ) at t and a point t′ ∈ X n = f −n(t). Since f n∶S2 ∖f −n(post(f )) → S2 ∖post(f ) is a covering map, can lift γ by f n starting at t′ to a curve ̃ γ. Set (t′)γ ∶= the endpoint of ̃ γ.
Tilings and IMG’s March 23, 2019 8 / 23
G ∶= π1(S2 ∖ post(f ),t) acts on the pre-image tree T by automorphisms via iterated monodromy action. γ ̃ γ t′ (t′)γ t Consider a loop γ ⊂ S2 ∖ post(f ) at t and a point t′ ∈ X n = f −n(t). Since f n∶S2 ∖f −n(post(f )) → S2 ∖post(f ) is 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] ∈ π1(S2 ∖ post(f ),t).
Tilings and IMG’s March 23, 2019 8 / 23
Formally, there is a group homomorphism ϕ∶G → Aut(T)
Tilings and IMG’s March 23, 2019 9 / 23
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).
Tilings and IMG’s March 23, 2019 9 / 23
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 natural labeling of T): for each g ∈ IMG(f ) and v ∈ T, g∣Tv ∶ Tv → Tvg corresponds to an element g∣v ∈ IMG(f ). This allows to encode the action of g on T using finite combinatorial data. Tvg Tv vg v g
Tilings and IMG’s March 23, 2019 9 / 23
1 a powerful invariant for pcf rational maps.
For instance, IMG(f ) recovers the Julia set Jf and f ∶ Jf → Jf .
Tilings and IMG’s March 23, 2019 10 / 23
1 a powerful invariant for pcf rational maps.
For instance, IMG(f ) recovers the Julia set Jf and f ∶ Jf → Jf .
2 a powerful algebraic tool to answer dynamical questions: ▸ Hubbard’s twisted rabbit problem; ▸ Global curve attractor problem.
Tilings and IMG’s March 23, 2019 10 / 23
1 a powerful invariant for pcf rational maps.
For instance, IMG(f ) recovers the Julia set Jf and f ∶ Jf → Jf .
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.
Tilings and IMG’s March 23, 2019 10 / 23
1 a powerful invariant for pcf rational maps.
For instance, IMG(f ) recovers the Julia set Jf and f ∶ Jf → Jf .
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(z2 + i); ▸ amenable groups of exponential growth, e.g., IMG(z2 − 1).
Tilings and IMG’s March 23, 2019 10 / 23
1 Main example and setup 2 Iterated monodromy groups 3 Growth of groups 4 Tiling approach to understanding of IMGs
Tilings and IMG’s March 23, 2019 11 / 23
Let G be a finitely generated group with a generating set S = {s1,...,sk}. Let ℓS(g) be the length of the element g ∈ G w.r.t. the set S, i.e., ℓS(g) ∶= min{n ∈ N0 ∶ g = sε1
1 ...sεn n , where sj ∈ S and εj ∈ {1,−1}}.
Tilings and IMG’s March 23, 2019 12 / 23
Let G be a finitely generated group with a generating set S = {s1,...,sk}. Let ℓS(g) be the length of the element g ∈ G w.r.t. the set S, i.e., ℓS(g) ∶= min{n ∈ N0 ∶ g = sε1
1 ...sεn n , where sj ∈ S and εj ∈ {1,−1}}.
The growth function of G w.r.t. S is defined by grG,S(n) = #{g ∈ G ∶ ℓS(g) ≤ n}.
Tilings and IMG’s March 23, 2019 12 / 23
Let G be a finitely generated group with a generating set S = {s1,...,sk}. Let ℓS(g) be the length of the element g ∈ G w.r.t. the set S, i.e., ℓS(g) ∶= min{n ∈ N0 ∶ g = sε1
1 ...sεn n , where sj ∈ S and εj ∈ {1,−1}}.
The growth function of G w.r.t. S is defined by grG,S(n) = #{g ∈ G ∶ ℓS(g) ≤ n}.
Definition
The group G is said to be of polynomial growth if grG,S(n) ≤ Cnk, for some C,k > 0. IMG(zd)
Tilings and IMG’s March 23, 2019 12 / 23
Let G be a finitely generated group with a generating set S = {s1,...,sk}. Let ℓS(g) be the length of the element g ∈ G w.r.t. the set S, i.e., ℓS(g) ∶= min{n ∈ N0 ∶ g = sε1
1 ...sεn n , where sj ∈ S and εj ∈ {1,−1}}.
The growth function of G w.r.t. S is defined by grG,S(n) = #{g ∈ G ∶ ℓS(g) ≤ n}.
Definition
The group G is said to be of polynomial growth if grG,S(n) ≤ Cnk, for some C,k > 0. IMG(zd) exponential growth if grG,S(n) ≥ ceα for some c,α > 0. IMG(z2 − 1)
Tilings and IMG’s March 23, 2019 12 / 23
Let G be a finitely generated group with a generating set S = {s1,...,sk}. Let ℓS(g) be the length of the element g ∈ G w.r.t. the set S, i.e., ℓS(g) ∶= min{n ∈ N0 ∶ g = sε1
1 ...sεn n , where sj ∈ S and εj ∈ {1,−1}}.
The growth function of G w.r.t. S is defined by grG,S(n) = #{g ∈ G ∶ ℓS(g) ≤ n}.
Definition
The group G is said to be of polynomial growth if grG,S(n) ≤ Cnk, for some C,k > 0. IMG(zd) exponential growth if grG,S(n) ≥ ceα for some c,α > 0. IMG(z2 − 1) intermediate growth, otherwise. IMG(z2 + i)
Tilings and IMG’s March 23, 2019 12 / 23
1 Main example and setup 2 Iterated monodromy groups 3 Growth of groups 4 Tiling approach to understanding of IMGs
Tilings and IMG’s March 23, 2019 13 / 23
Let f be a Thurston map with an invariant graph G ⊃ post(f ). The preimages f −n(G), n ∈ N, define subdivisions of the sphere S2, called tilings. Combinatorial properties of tilings ← → Algebraic properties of IMG’s (growth, torsion, amenability?)
Tilings and IMG’s March 23, 2019 14 / 23
Fix a basepoint t ∈ ̂ C ∖ post(f ), such that t is in the white 0-tile T. Identify X n ∶= f −n(t) with {white n-tiles, i.e., components of f −n(T)}. Every white n-tile contains a point from f −n(t); every point t′ ∈ f −n(t) is contained in a white n-tile. t b
1
a
∞
c
−1
T Fix loops a,b,c at t around the postcritical points. a,b,c generate π1(̂ C ∖ post(f ),t), acb = 1.
Tilings and IMG’s March 23, 2019 15 / 23
Let v ∈ f −n(post(f )) be an n-vertex. The n-flower of v is W n(v) ∶= ⋃{n-tile T′ ∶ v ∈ T′}.
Tilings and IMG’s March 23, 2019 16 / 23
Let v ∈ f −n(post(f )) be an n-vertex. The n-flower of v is W n(v) ∶= ⋃{n-tile T′ ∶ v ∈ T′}. degree is deg(f n,v), i.e., the number of white (and black) n-tiles composing W n(v). type a,b,c (or red, blue, green), if f n(v) = ∞,1,−1 (i.e., the color of the 0-vertex f n(v)).
Tilings and IMG’s March 23, 2019 16 / 23
Let v ∈ f −n(post(f )) be an n-vertex. The n-flower of v is W n(v) ∶= ⋃{n-tile T′ ∶ v ∈ T′}. degree is deg(f n,v), i.e., the number of white (and black) n-tiles composing W n(v). type a,b,c (or red, blue, green), if f n(v) = ∞,1,−1 (i.e., the color of the 0-vertex f n(v)). flower of degree 8 and of type b b acts by rotating tiles around the center in a b-flower every n-tile is contained in one n-flower of type a,b,c
Tilings and IMG’s March 23, 2019 16 / 23
b 1 a ∞ ∞ c −1 1 −1 ∞ ∞ a0 a0 a1 a1 c0 c1 Attention: generator b rotates all b-flowers at the same time. Same for a and c.
Tilings and IMG’s March 23, 2019 17 / 23
b 1 a ∞ ∞ c −1 1 −1 ∞ ∞ a0 a0 a1 a1 c0 c1 Attention: generator b rotates all b-flowers at the same time. Same for a and c. What are the orders of the generators a, b, c?
Tilings and IMG’s March 23, 2019 17 / 23
b 1 a ∞ ∞ c −1 1 −1 ∞ ∞ a0 a0 a1 a1 c0 c1 Attention: generator b rotates all b-flowers at the same time. Same for a and c. What are the orders of the generators a, b, c?
c0
3∶1
2∶1
1
4∶1
3∶1
2∶1
Similarly, ord(c) = 3, ord(b) = 24.
Tilings and IMG’s March 23, 2019 17 / 23
Theorem (H.-Meyer)
The IMG of the triangular snowball map f is of exponential growth.
Tilings and IMG’s March 23, 2019 18 / 23
Theorem (H.-Meyer)
The IMG of the triangular snowball map f is of exponential growth. Main idea: construct a free semigroup in IMG(f ).
Tilings and IMG’s March 23, 2019 18 / 23
The 0-edge [1,∞] is invariant under f . In fact, f ∶[1,∞] → [1,∞] is 2 ∶ 1.
Tilings and IMG’s March 23, 2019 19 / 23
The 0-edge [1,∞] is invariant under f . In fact, f ∶[1,∞] → [1,∞] is 2 ∶ 1. 1 ⋯ ∞ b-flowers on [1,∞] are of degree 8, a-flowers are of degree 2.
Tilings and IMG’s March 23, 2019 19 / 23
The 0-edge [1,∞] is invariant under f . In fact, f ∶[1,∞] → [1,∞] is 2 ∶ 1. 1 ⋯ ∞ b-flowers on [1,∞] are of degree 8, a-flowers are of degree 2. Therefore, ab4 acts by “shifting tiles to the right” on [1,∞]. Similarly, abk1abk2 ...abkn, where kj = 4,12,20, shift tiles to the right n times.
Tilings and IMG’s March 23, 2019 19 / 23
The 0-edge [1,∞] is invariant under f . In fact, f ∶[1,∞] → [1,∞] is 2 ∶ 1. 1 ⋯ ∞ b-flowers on [1,∞] are of degree 8, a-flowers are of degree 2. Therefore, ab4 acts by “shifting tiles to the right” on [1,∞]. Similarly, abk1abk2 ...abkn, where kj = 4,12,20, shift tiles to the right n times.
Claim
Elements ab4,ab12,ab20 are of infinite order in IMG(f ).
Tilings and IMG’s March 23, 2019 19 / 23
Claim
Elements ab4,ab12,ab20 generate a free semigroup in IMG(f ).
Tilings and IMG’s March 23, 2019 20 / 23
Claim
Elements ab4,ab12,ab20 generate a free semigroup in IMG(f ). Enough to show that bk1abk2 ...abkn ≠ bj1abj2 ...abjn in IMG(f ) for k1 ≠ j1.
Tilings and IMG’s March 23, 2019 20 / 23
Claim
Elements ab4,ab12,ab20 generate a free semigroup in IMG(f ). Enough to show that bk1abk2 ...abkn ≠ bj1abj2 ...abjn in IMG(f ) for k1 ≠ j1. The point c0 satisfies f 2(c0) = 1 and deg(f 2,c0) = 3.
c0
Tilings and IMG’s March 23, 2019 20 / 23
Claim
Elements ab4,ab12,ab20 generate a free semigroup in IMG(f ). Enough to show that bk1abk2 ...abkn ≠ bj1abj2 ...abjn in IMG(f ) for k1 ≠ j1. The point c0 satisfies f 2(c0) = 1 and deg(f 2,c0) = 3.
c0
bk1 and bj1 map a white tile T′ ∋ c0 to distinct tiles (deg(f 2,c0) = 3 ∤ k1 − j1). The rest of the words shift the tiles along distinct preimages of [1,∞].
Tilings and IMG’s March 23, 2019 20 / 23
Theorem (H.-Meyer)
There exists a non-renormalizable pcf rational map f ∶ ̂ C → ̂ C (in fact, an infinite family of maps) with IMG(f ) having exponential growth, s.t., one
1 Jf = ̂
C;
2 Jf ≅ Sierpi´
nski carpet;
3 Jf is a dendrite.
Tilings and IMG’s March 23, 2019 21 / 23
Dynamical properties of f , ← → Algebraic properties of IMG(f) properties of tilings (growth, torsion, amenability)
Tilings and IMG’s March 23, 2019 22 / 23
Dynamical properties of f , ← → Algebraic properties of IMG(f) properties of tilings (growth, torsion, amenability)
1 Which rational maps have IMG’s of intermediate growth?
Tilings and IMG’s March 23, 2019 22 / 23
Dynamical properties of f , ← → Algebraic properties of IMG(f) properties of tilings (growth, torsion, amenability)
1 Which rational maps have IMG’s of intermediate growth? 2 Are there rational maps whose IMG’s are torsion?
Tilings and IMG’s March 23, 2019 22 / 23
Dynamical properties of f , ← → Algebraic properties of IMG(f) properties of tilings (growth, torsion, amenability)
1 Which rational maps have IMG’s of intermediate growth? 2 Are there rational maps whose IMG’s are torsion? 3 Which rational maps have amenabe IMG’s?
Tilings and IMG’s March 23, 2019 22 / 23
THANK YOU!
Tilings and IMG’s March 23, 2019 23 / 23