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

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

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

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

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

On Lebesgue measure of integral self-affine sets

Let $A$ be an expanding integer $n\times n$ matrix and $D$ be a finite subset of $Z^n$. The self-affine set $T=T(A,D)$ is the unique compact set satisfying the equality $A(T)=\cup_{d\in D} (T+d)$. We present an effective algorithm to compute the Lebesgue measure of the self-affine set $T$, the measure of intersection $T\cap (T+u)$ for $u\in Z^n$, and the measure of intersection of self-affine sets $T(A,D_1)\cap T(A,D_2)$ for different sets $D_1,D_2\subset Z^n$.Comment: 5 pages, 1 figur

Trends in the Development of University Education in the Postmodern Period

The purpose of the research was to study the changes in the ranking of professional skills of students, which determine trends in university education in the postmodern era. The presented results of the experiment allowed to determine the level of perception of new professional skills by university students under the influence of the ideology of postmodernism. The general hypothesis of the study was that postmodern ideology influences the model of university education, contributes to the creation of the foundations for the formation of new professional skills of students. This study is part of a broader study that explores ways to improve the quality of university education based on the substantive professional priorities of students in the context of postmodernism. The results of the experiment allowed us to draw conclusions about the relationship between the quality of education and professional priorities of students, a properly organized system of university education in the postmodern period

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