GROUPS ACTING ON 1-DIMENSIONAL SPACES AND SOME QUESTIONS Andr´esNavas Universidad de Santiago de Chile ICM Special Session \Dynamical Systems and Ordinary Differential Equations" Rio de Janeiro, August 2018 A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Thinking from a group-theoretical perspec- tive can also give useful insight into \more classical" dynamics. A General Principle If a group (class of groups) acts nicely on a nice space, then the action should reveal so- me algebraic structure ( nice theorem). A General Principle If a group (class of groups) acts nicely on a nice space, then the action should reveal so- me algebraic structure ( nice theorem). Thinking from a group-theoretical perspec- tive can also give useful insight into \more classical" dynamics. The group of piecewise dyadic homeomorphisms of the binary Cantor• set is finitely presented and simple (Thompson's group V ). It contains a copy of every finite group. • Its automorphism group is large but can been understood dynamically• (Bleak, Cameron, Lanoue, Yonah, N). Groups acting on the Cantor set Every group Γ acts on 0; 1 Γ by shifting coordinates. For infinite countable Γ, this is a Cantorf g set. Which groups can arise as groups of isometries (diffeomorphisms) of a Cantor set ? The group of piecewise dyadic homeomorphisms of the binary Cantor• set is finitely presented and simple (Thompson's group V ). It contains a copy of every finite group. • Its automorphism group is large but can been understood dynamically• (Bleak, Cameron, Lanoue, Yonah, N). Groups acting on the Cantor set Every group Γ acts on 0; 1 Γ by shifting coordinates. For infinite countable Γ, this is a Cantorf g set. Which groups can arise as groups of isometries (diffeomorphisms) of a Cantor set ? Groups acting on the Cantor set Every group Γ acts on 0; 1 Γ by shifting coordinates. For infinite countable Γ, this is a Cantorf g set. Which groups can arise as groups of isometries (diffeomorphisms) of a Cantor set ? The group of piecewise dyadic homeomorphisms of the binary Cantor• set is finitely presented and simple (Thompson's group V ). It contains a copy of every finite group. • Its automorphism group is large but can been understood dynamically• (Bleak, Cameron, Lanoue, Yonah, N). The elements of T that respect the linear order form the sub- group• F ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the unit interval. F = f ; g :[fg −1; f −1gf ] = [fg −1; f −2gf 2] = id : ... ... 1 .... 1 .... .... .... .... ... .... .... .... .... .... .... .... ....... .... 3/4 ....... .... ....... .... ...... .... 5/8 ............ .... .......... .... .......... ..... ......... 1/2 ....... 1/2 ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... f ...... g ........ ....... 1/4 .......... ....... .......... ...... .......... ....... .......... ....... .......... ...... .......... ....... 0 ......... 0 ...... 1 3 1 3 7 0 2 4 1 0 2 4 8 1 Is Thompson's group F amenable ? A remarkable group closely related to F recently constructed and proven to be non-amenable by Justin-Moore and Lodha. 1 Richard Thompson's groups The elements of V that respect the cyclic order form the sub- group• T ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the circle. F = f ; g :[fg −1; f −1gf ] = [fg −1; f −2gf 2] = id : ... ... 1 .... 1 .... .... .... .... ... .... .... .... .... .... .... .... ....... .... 3/4 ....... .... ....... .... ...... .... 5/8 ............ .... .......... .... .......... ..... ......... 1/2 ....... 1/2 ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... f ...... g ........ ....... 1/4 .......... ....... .......... ...... .......... ....... .......... ....... .......... ...... .......... ....... 0 ......... 0 ...... 1 3 1 3 7 0 2 4 1 0 2 4 8 1 Is Thompson's group F amenable ? A remarkable group closely related to F recently constructed and proven to be non-amenable by Justin-Moore and Lodha. 1 Richard Thompson's groups The elements of V that respect the cyclic order form the sub- group• T ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the circle. The elements of T that respect the linear order form the sub- group• F ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the unit interval. ... ... 1 .... 1 .... .... .... .... ... .... .... .... .... .... .... .... ....... .... 3/4 ....... .... ....... .... ...... .... 5/8 ............ .... .......... .... .......... ..... ......... 1/2 ....... 1/2 ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... f ...... g ........ ....... 1/4 .......... ....... .......... ...... .......... ....... .......... ....... .......... ...... .......... ....... 0 ......... 0 ...... 1 3 1 3 7 0 2 4 1 0 2 4 8 1 Is Thompson's group F amenable ? A remarkable group closely related to F recently constructed and proven to be non-amenable by Justin-Moore and Lodha. 1 Richard Thompson's groups The elements of V that respect the cyclic order form the sub- group• T ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the circle. The elements of T that respect the linear order form the sub- group• F ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the unit interval. F = f ; g :[fg −1; f −1gf ] = [fg −1; f −2gf 2] = id : ... ... 1 .... 1 .... .... .... .... ... .... .... .... .... .... .... .... ....... .... 3/4 ....... .... ....... .... ...... .... 5/8 ............ .... .......... .... .......... ..... ......... 1/2 ....... 1/2 ....... ....... ....... ....... ....... ....... ....... ...... ....... ....... f ...... g ........ ....... 1/4 .......... ....... .......... ...... .......... ....... .......... ....... .......... ...... .......... ....... 0 ......... 0 ...... 1 3 1 3 7 0 2 4 1 0 2 4 8 1 1 Richard Thompson's groups The elements of V that respect the cyclic order form the sub- group• T ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the circle. The elements of T that respect the linear order form the sub- group• F ; this may be seen also as a group of piecewise-affine, orientation-preserving homeomorphisms of the unit interval. F = f ; g :[fg −1; f −1gf ] = [fg −1; f −2gf 2] = id : Is Thompson's group F amenable ? A remarkable group closely related to F recently constructed and proven to be non-amenable by Justin-Moore and Lodha. YES for linear groups (Burnside). • The Burnside groups: • B(n) := a; b : w n = id for all w h i - B(2); B(3); B(4) and B(6) are finite. - B(7) should be infinite (hyperbolic; obvious for Gromov). - B(5) should still be infinite (Zelmanov). - For odd n>666, B(n) is infinite (non-amenable; Adian-Novikov). The Burnside groups Let Γ be a finitely-generated group in which every element has finite order. Is Γ necessarily finite ? [Burnside] The Burnside groups: • B(n) := a; b : w n = id for all w h i - B(2); B(3); B(4) and B(6) are finite. - B(7) should be infinite (hyperbolic; obvious for Gromov). - B(5) should still be infinite (Zelmanov). - For odd n>666, B(n) is infinite (non-amenable; Adian-Novikov). The Burnside groups Let Γ be a finitely-generated group in which every element has finite order. Is Γ necessarily finite ? [Burnside]