52 research outputs found
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 to every covering of a topological space
by its open subset. It is the quotient of the fundamental group
by the intersection of the kernels of its monodromy action for the iterates
. 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 is related to the group. In particular, the
Julia set of can be reconstructed from \img(f) (from its action on the
tree), if 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 and simple
finitely generated algebras of quadratic growth over arbitrary fields.Comment: 19 page
Mating, paper folding, and an endomorphism of PC^2
We are studying topological properties of the Julia set of the map of the complex projective plane 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
Finitely presented groups associated with expanding maps
We associate with every locally expanding self-covering of a
compact path connected metric space a finitely presented group . We prove
that this group is a complete invariant of the dynamical system: two groups
and are isomorphic as abstract groups if and only if the
corresponding dynamical systems are topologically conjugate. We also show that
the commutator subgroup of is simple, and give a topological
interpretation of .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
The Julia set of a post-critically finite endomorphism of PC^2
We construct a combinatorial model of the Julia set of the endomorphism of .Comment: 60 pages, 28 figure
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