Palindromic subshifts and simple periodic groups of intermediate growth

We describe a new class of groups of Burnside type, giving a procedure transforming an arbitrary non-free minimal action of the dihedral group on a Cantor set into an orbit-equivalent action of an infinite finitely generated periodic group. We show that if the associated Schreier graphs are linearly repetitive, then the group is of intermediate growth. In particular, this gives first examples of simple groups of intermediate growth.Comment: 40 pages, 12 figure

Simple groups of dynamical origin

We associate with every etale groupoid G two normal subgroups S(G) and A(G) of the topological full group of G, which are analogs of the symmetric and alternating groups. We prove that if G is a minimal groupoid of germs (e.g., of a group action), then A(G) is simple and is contained in every non-trivial normal subgroup of the full group. We show that if G is expansive (e.g., is the groupoid of germs of an expansive action of a group), then A(G) is finitely generated. We also show that S(G)/A(G) is a quotient of H_0(G, Z/2Z).Comment: 25 pages, 3 figure

Hyperbolic groupoids: metric and measure

We construct Patterson-Sullivan measure and a natural metric on the unit space of a hyperbolic groupoid. In particular, this gives a new approach to defining SRB measures on Smale spaces using Gromov hyperbolic graphs.Comment: 36 page

Iterated Monodromy Groups

We associate a group $IMG(f)$ to every covering $f$ of a topological space $M$ by its open subset. It is the quotient of the fundamental group $\pi_1(M)$ by the intersection of the kernels of its monodromy action for the iterates $f^n$. Every iterated monodromy group comes together with a naturally defined action on a rooted tree. We present an effective method to compute this action and show how the dynamics of $f$ is related to the group. In particular, the Julia set of $f$ can be reconstructed from \img(f) (from its action on the tree), if $f$ is expanding.Comment: about 40 pages, 6 figure

An uncountable family of 3-generated groups with isomorphic profinite completions

We construct an uncountable family of 3-generated residually finite just-infinite groups with isomorphic profinite completions. We also show that word growth rate is not a profinite property.Comment: 12 pages, 3 figure

Growth of etale groupoids and simple algebras

We study growth and complexity of \'etale groupoids in relation to growth of their convolution algebras. As an application, we construct simple finitely generated algebras of arbitrary Gelfand-Kirillov dimension $\ge 2$ and simple finitely generated algebras of quadratic growth over arbitrary fields.Comment: 19 page

Finitely presented groups associated with expanding maps

We associate with every locally expanding self-covering $f:M\to M$ of a compact path connected metric space a finitely presented group $V_f$. We prove that this group is a complete invariant of the dynamical system: two groups $V_{f_1}$ and $V_{f_2}$ are isomorphic as abstract groups if and only if the corresponding dynamical systems are topologically conjugate. We also show that the commutator subgroup of $V_f$ is simple, and give a topological interpretation of $V_f/V_f'$.Comment: 46 pages, 13 figure

Locally connected Smale spaces, pinched spectrum, and infra-nilmanifolds

We show that if (X, f) is a locally connected Smale space such that the local product structure on X can be lifted by a covering with virtually nilpotent group of deck transformations to a global direct product, then (X, f) is topologically conjugate to a hyperbolic infra-nilmanifold automorphism. We use this result to give a generalization to Smale spaces of a theorem of M. Brin and A. Manning on Anosov diffeomorphisms with pinched spectrum, and to show that every locally connected codimension one Smale space is topologically conjugate to a hyperbolic automorphism of a torus.Comment: 55 pages, 5 figure

Mating, paper folding, and an endomorphism of PC^2

We are studying topological properties of the Julia set of the map $F(z, p)=((2z/(p+1)-1)^2, ((p-1)/(p+1))^2)$ of the complex projective plane $PC^2$ to itself. We show a relation of this rational function with an uncountable family of "paper folding" plane filling curves.Comment: 53 pages, 36 figure

The Julia set of a post-critically finite endomorphism of PC^2

We construct a combinatorial model of the Julia set of the endomorphism $f(z, w)=((1-2z/w)^2, (1-2/w)^2)$ of $PC^2$.Comment: 60 pages, 28 figure