Iterated Minkowski sums, horoballs and north-south dynamics - PowerPoint PPT Presentation

Iterated Minkowski sums, horoballs and north-south dynamics Jeremias Epperlein (joint with Tom Meyerovitch) Ben-Gurion University of the Negev 05.06.2020 1 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } M = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 1 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 2 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 3 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 4 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 5 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 6 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 7 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 8 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 9 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 10 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 11 A ( M ) = A = 2 / 13

Iterated Minkowski sums Setting » G finitely generated group » A ∋ 1 G generates G as a semigroup, i.e. G = ⋃ ∞ n = 0 A n » X = 2 G ≅ { 0 , 1 } G with product topology » ϕ A ∶ X → X, ϕ A ( M ) = MA = { ma ∶ m ∈ M,a ∈ A } ϕ 12 A ( M ) = A = 2 / 13

Invariant properties

Invariant properties Call a property P of ( G,A ) dynamically recognizable (among a set of groups G ) if for all groups G 1 ,G 2 (for all groups in G 1 ,G 2 ∈ G ) and positive generating sets A 1 ,A 2 (( G 1 ,A 1 ) has P ∧ ( 2 G 1 ,ϕ A 1 ) ≅ ( 2 G 2 ,ϕ A 2 )) � ⇒ ( G 2 ,A 2 ) 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 = { Z d ∶ d ∈ N } . 4 / 13

Counting preimages Key idea » L k A ( M )}∣ ≅ ∣ M ∣ . A ( M ) ∶= log 2 ∣{ N ⊆ G ∶ ϕ k A ( N ) = ϕ k » Study the growth of n ↦ L k ( n ) A ( M )) . ( ϕ n A Example We can characterize the finite subsets of G : Fin ( ϕ A ) = { M ∈ X ∶ ∣ ϕ − r A ({ ϕ r + n A ( M )})∣ < ∞ ∀ r,n ∈ N } . 5 / 13

Amenability Theorem Let k ∈ N be such that A − 1 ⊆ ϕ k A ({ 1 G }) . Then G is amenable iff there is a sequence of sets ( M n ) n ∈ N in Fin ( ϕ A ) such that A ( ϕ ( k + 5 ) k L k ( M n )) A = 1 . lim A ( ϕ ( k + 1 ) k n → ∞ ( M n )) L k A where L k A ( M )}∣ . A ( M ) ∶= log 2 ∣{ N ⊆ G ∶ ϕ k A ( N ) = ϕ k 6 / 13

Horoballs and the eventual image

Horoballs I Definition The balls B G = { gA n ∶ v ∈ G,n ∈ N } form a forward invariant subsystem. The new sets in the closure B G ∖ (B G ∪ {∅ ,G }) are called horoballs . , , , ,... → Proposition The eventual image ⋂ ∞ n = 0 ϕ A ( X ) of ϕ A consists of all unions of horoballs. 8 / 13

Horoballs II Example we have 12 horoballs up to translation. For A = 9 / 13

The natural extension

North-South dynamics Definition (Natural extension) X = {( x i ) i ∈ Z ∈ X Z ∶ x i + 1 = ϕ A ( x i )} ⊆ X Z , » ˆ » ˆ ϕ A (( x j ) j ∈ Z ) i = ϕ A ( x i ) = x i + 1 . The natural extension of ϕ A exhibits north-south dynamics . » lim n →∞ ˆ ϕ n A ( x ) = 1 , » lim n →∞ ˆ ϕ − n A ( x ) = 0 . Theorem For G = Z d ,d ≥ 2 , the natural extension ˆ X of ϕ A is a Cantor space. The natural extensions of ( Z d 1 ,ϕ A 1 ) and ( Z d 2 ,ϕ A 2 ) , d 1 ,d 2 ≥ 2 are topologically conjugate for all A 1 ,A 2 . Remark The topological structure of the eventual image ⋂ n ϕ n G ( X ) is more complicated and depends on the geometry of the generating set. 11 / 13

Perfectness of the natural extension Horoballs in Z d In Z d 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 12 / 13

Factoring Question When does ( 2 G ,ϕ A ) factor onto ( 2 H ,ϕ B ) ? Does ( 2 Z 2 ,ϕ { − 1 , 0 , 1 } 2 ) factor onto ( 2 Z ,ϕ { − 1 , 0 , 1 } ) ? What we know: » ( 2 Z 2 ,ϕ { − 1 , 0 , 1 } 2 ) factors onto ( 2 N ,ϕ { − 1 , 0 , 1 } ) . » ( 2 Z ,ϕ { − 1 , 0 , 1 } ) does not factor onto ( 2 Z 2 ,ϕ { − 1 , 0 , 1 } 2 ) . » EI ( 2 Z 2 ,ϕ { − 1 , 0 , 1 } 2 ) factors onto EI ( 2 Z ,ϕ { − 1 , 0 , 1 } ) (where EI dentotes the restriction to the eventual image). 13 / 13