53 research outputs found
Iterated Monodromy Groups of Quadratic Polynomials, I
We describe the iterated monodromy groups associated with post-critically
finite quadratic polynomials, and explicit their connection to the `kneading
sequence' of the polynomial.
We then give recursive presentations by generators and relations for these
groups, and study some of their properties, like torsion and `branchness'.Comment: 18 pages, 3 EPS figure
Virtual endomorphisms of groups
Dedicated to V. V. Kirichenko on the occasion of his 60th birthda
Hyperbolic spaces from self-similar group actions
We show that the limit space of a contracting selfsimilar group action is the boundary of a naturally defined Gromov
hyperbolic space
Automata, Groups, Limit Spaces, and Tilings
We explore the connections between automata, groups, limit spaces of
self-similar actions, and tilings. In particular, we show how a group acting
``nicely'' on a tree gives rise to a self-covering of a topological groupoid,
and how the group can be reconstructed from the groupoid and its covering. The
connection is via finite-state automata. These define decomposition rules, or
self-similar tilings, on leaves of the solenoid associated with the covering.Comment: to appear in J. Algebr
Post-critically finite self-similar groups
We describe in terms of automata theory the automatic actions with post-critically finite limit space. We prove that
these actions are precisely the actions by bounded automata and
that any self-similar action by bounded automata is contracting
On Lebesgue measure of integral self-affine sets
Let be an expanding integer matrix and be a finite subset
of . The self-affine set is the unique compact set satisfying
the equality . We present an effective algorithm to
compute the Lebesgue measure of the self-affine set , the measure of
intersection for , and the measure of intersection of
self-affine sets for different sets .Comment: 5 pages, 1 figur
The tight groupoid of an inverse semigroup
In this work we present algebraic conditions on an inverse semigroup S (with zero) which imply that its associated tight groupoid G_tight(S) is: Hausdorff, essentially principal, minimal and contracting, respectively. In some cases these conditions are in fact necessary and sufficient.The first-named author was partially supported by CNPq. The second-named author was partially supported by PAI III grants FQM-298 and P11-FQM-7156 of the Junta de Andalucía and by the DGI- MICINN and European Regional Development Fund, jointly, through Project MTM2011-28992-C02-02
Finite self-similar p-groups with abelian first level stabilizers
We determine all finite p-groups that admit a faithful, self-similar action
on the p-ary rooted tree such that the first level stabilizer is abelian. A
group is in this class if and only if it is a split extension of an elementary
abelian p-group by a cyclic group of order p.
The proof is based on use of virtual endomorphisms. In this context the
result says that if G is a finite p-group with abelian subgroup H of index p,
then there exists a virtual endomorphism of G with trivial core and domain H if
and only if G is a split extension of H and H is an elementary abelian p-group.Comment: one direction of theorem 2 extended to regular p-group
A characterization of those automata that structurally generate finite groups
Antonenko and Russyev independently have shown that any Mealy automaton with
no cycles with exit--that is, where every cycle in the underlying directed
graph is a sink component--generates a fi- nite (semi)group, regardless of the
choice of the production functions. Antonenko has proved that this constitutes
a characterization in the non-invertible case and asked for the invertible
case, which is proved in this paper
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