Words in non-periodic branch groups Introduction Groups acting on - PowerPoint PPT Presentation

Words in non-periodic branch groups Elisabeth Fink Words in non-periodic branch groups Introduction Groups acting on Rooted Trees Elisabeth Fink Rooted Trees Automorphisms A University of Oxford Construction Construction Words May 28, 2013 An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Introduction Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Introduction Words in non-periodic branch groups Elisabeth Fink A construction of a branch group G with Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Introduction Words in non-periodic branch groups Elisabeth Fink A construction of a branch group G with Introduction no non-abelian free subgroups Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Introduction Words in non-periodic branch groups Elisabeth Fink A construction of a branch group G with Introduction no non-abelian free subgroups Groups acting on exponential growth Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Introduction Words in non-periodic branch groups Elisabeth Fink A construction of a branch group G with Introduction no non-abelian free subgroups Groups acting on exponential growth Rooted Trees Rooted Trees For any g , h ∈ G we construct w g , h ( x , y ) ∈ F ( x , y ) with Automorphisms A w g , h ( g , h ) = 1 ∈ G . Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Rooted Trees Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Rooted Trees Words in Rooted tree: cyclefree graph with vertices V and edges E non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Rooted Trees Words in Rooted tree: cyclefree graph with vertices V and edges E non-periodic branch distinguished root: r groups Elisabeth Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Rooted Trees Words in Rooted tree: cyclefree graph with vertices V and edges E non-periodic branch distinguished root: r groups distance function: d ( v , w ) , number of edges in unique path from v Elisabeth to w Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Rooted Trees Words in Rooted tree: cyclefree graph with vertices V and edges E non-periodic branch distinguished root: r groups distance function: d ( v , w ) , number of edges in unique path from v Elisabeth to w Fink level n : Ω( n ) = { v ∈ V : d ( v , r ) = n } Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Rooted Trees Words in Rooted tree: cyclefree graph with vertices V and edges E non-periodic branch distinguished root: r groups distance function: d ( v , w ) , number of edges in unique path from v Elisabeth to w Fink level n : Ω( n ) = { v ∈ V : d ( v , r ) = n } Introduction Groups T n = ( V ′ , E ′ ) with acting on Rooted Trees V ′ = { v ∈ V : d ( v , r ) ≤ n } Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Rooted Trees Words in Rooted tree: cyclefree graph with vertices V and edges E non-periodic branch distinguished root: r groups distance function: d ( v , w ) , number of edges in unique path from v Elisabeth to w Fink level n : Ω( n ) = { v ∈ V : d ( v , r ) = n } Introduction Groups T n = ( V ′ , E ′ ) with acting on Rooted Trees V ′ = { v ∈ V : d ( v , r ) ≤ n } Rooted Trees Automorphisms E ′ = { e ∈ E : e = e vw , v , w ∈ V ′ } A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Rooted Trees Words in Rooted tree: cyclefree graph with vertices V and edges E non-periodic branch distinguished root: r groups distance function: d ( v , w ) , number of edges in unique path from v Elisabeth to w Fink level n : Ω( n ) = { v ∈ V : d ( v , r ) = n } Introduction Groups T n = ( V ′ , E ′ ) with acting on Rooted Trees V ′ = { v ∈ V : d ( v , r ) ≤ n } Rooted Trees Automorphisms E ′ = { e ∈ E : e = e vw , v , w ∈ V ′ } A Construction T v : subtree with root v ∈ V Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Rooted Trees Words in Rooted tree: cyclefree graph with vertices V and edges E non-periodic branch distinguished root: r groups distance function: d ( v , w ) , number of edges in unique path from v Elisabeth to w Fink level n : Ω( n ) = { v ∈ V : d ( v , r ) = n } Introduction Groups T n = ( V ′ , E ′ ) with acting on Rooted Trees V ′ = { v ∈ V : d ( v , r ) ≤ n } Rooted Trees Automorphisms E ′ = { e ∈ E : e = e vw , v , w ∈ V ′ } A Construction T v : subtree with root v ∈ V Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Automorphisms acting on rooted trees Words in non-periodic branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Automorphisms acting on rooted trees Words in non-periodic acting on vertices, preserve edge incidence and root branch groups Elisabeth Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Automorphisms acting on rooted trees Words in non-periodic acting on vertices, preserve edge incidence and root branch groups st G ( n ) = { g ∈ G : g acts trivially on T n } Elisabeth Fink Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Automorphisms acting on rooted trees Words in non-periodic acting on vertices, preserve edge incidence and root branch groups st G ( n ) = { g ∈ G : g acts trivially on T n } Elisabeth Fink rst G ( v ) = { g ∈ G : g acts trivially on T \ T v } Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Automorphisms acting on rooted trees Words in non-periodic acting on vertices, preserve edge incidence and root branch groups st G ( n ) = { g ∈ G : g acts trivially on T n } Elisabeth Fink rst G ( v ) = { g ∈ G : g acts trivially on T \ T v } Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups

Automorphisms acting on rooted trees Words in non-periodic acting on vertices, preserve edge incidence and root branch groups st G ( n ) = { g ∈ G : g acts trivially on T n } Elisabeth Fink rst G ( v ) = { g ∈ G : g acts trivially on T \ T v } Introduction Groups acting on Rooted Trees Rooted Trees Automorphisms A Construction Construction Words An Example Lemmata and a Theorem Growth Further Questions Elisabeth Fink Words in non-periodic branch groups