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Iterated Minkowski sums, horoballs and north-south dynamics

Jeremias Epperlein (joint with Tom Meyerovitch)

Ben-Gurion University of the Negev

05.06.2020

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Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A}

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Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} M = A =

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Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ1

A(M) =

A =

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Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ2

A(M) =

A =

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Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ3

A(M) =

A =

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Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ4

A(M) =

A =

2 / 13

Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ5

A(M) =

A =

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Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ6

A(M) =

A =

2 / 13

Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ7

A(M) =

A =

2 / 13

Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ8

A(M) =

A =

2 / 13

Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ9

A(M) =

A =

2 / 13

Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ10

A (M) =

A =

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Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ11

A (M) =

A =

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Iterated Minkowski sums Setting

» G finitely generated group » A ∋ 1G generates G as a semigroup, i.e. G = ⋃∞

n=0 An

» X = 2G ≅ {0,1}G with product topology » ϕA ∶ X → X, ϕA(M) = MA = {ma ∶ m ∈ M,a ∈ A} ϕ12

A (M) =

A =

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Invariant properties

Invariant properties

Call a property P of (G,A) dynamically recognizable (among a set of groups G) if for all groups G1,G2 (for all groups in G1,G2 ∈ G) and positive generating sets A1,A2 ((G1,A1) has P ∧ (2G1,ϕA1) ≅ (2G2,ϕA2)) ⇒ (G2,A2) has P.

Theorem

The following properties are dynamically recognizable: » amenability, » the growth type (exponential, polynomial, ... ), » the exponential growth rate among e.g. free groups, » rank and vol(conv(A)) among G = {Zd ∶ d ∈ N}.

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Counting preimages Key idea

» Lk

A(M) ∶= log2 ∣{N ⊆ G ∶ ϕk A(N) = ϕk A(M)}∣ ≅ ∣M∣.

» Study the growth of n ↦ Lk(n)

A

(ϕn

A(M)).

Example

We can characterize the finite subsets of G: Fin(ϕA) = {M ∈ X ∶ ∣ϕ−r

A ({ϕr+n A (M)})∣ < ∞ ∀r,n ∈ N}.

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Amenability Theorem

Let k ∈ N be such that A−1 ⊆ ϕk

A({1G}). Then G is amenable iff

there is a sequence of sets (Mn)n∈N in Fin(ϕA) such that lim

n→∞

Lk

A(ϕ(k+5)k A

(Mn)) Lk

A(ϕ(k+1)k A

(Mn)) = 1. where Lk

A(M) ∶= log2 ∣{N ⊆ G ∶ ϕk A(N) = ϕk A(M)}∣.

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Horoballs and the eventual image

Horoballs I Definition

The balls BG = {gAn ∶ v ∈ G,n ∈ N} form a forward invariant

• subsystem. The new sets in the closure BG ∖ (BG ∪ {∅,G}) are

called horoballs. , , , ,... →

Proposition

The eventual image ⋂∞

n=0 ϕA(X) of ϕA consists of all unions of

horoballs.

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Horoballs II Example

For A = we have 12 horoballs up to translation.

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The natural extension

North-South dynamics Definition (Natural extension)

» ˆ X = {(xi)i∈Z ∈ XZ ∶ xi+1 = ϕA(xi)} ⊆ XZ, » ˆ ϕA((xj)j∈Z)i = ϕA(xi) = xi+1. The natural extension of ϕA exhibits north-south dynamics. » limn→∞ ˆ ϕn

A(x) = 1,

» limn→∞ ˆ ϕ−n

A (x) = 0.

Theorem

For G = Zd,d ≥ 2, the natural extension ˆ X of ϕA is a Cantor

• space. The natural extensions of (Zd1,ϕA1) and (Zd2,ϕA2),

d1,d2 ≥ 2 are topologically conjugate for all A1,A2.

Remark

The topological structure of the eventual image ⋂n ϕn

G(X) is

more complicated and depends on the geometry of the generating set.

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Perfectness of the natural extension Horoballs in Zd

In Zd we have, » a one-to-one correspondence between faces of conv(A) and A-horoballs, » finite unions of vertex-horoballs are dense.

Travel back in time and zoom out

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Factoring Question

When does (2G,ϕA) factor onto (2H,ϕB)? Does (2Z2,ϕ{−1,0,1}2) factor onto (2Z,ϕ{−1,0,1})?

What we know:

» (2Z2,ϕ{−1,0,1}2) factors onto (2N,ϕ{−1,0,1}). » (2Z,ϕ{−1,0,1}) does not factor onto (2Z2,ϕ{−1,0,1}2). » EI(2Z2,ϕ{−1,0,1}2) factors onto EI(2Z,ϕ{−1,0,1}) (where EI dentotes the restriction to the eventual image).

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