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Zoran Sunic

Publications and source records attributed to Zoran Sunic.

34 records · Page 2Linked to original sources
arXiv2010

Pattern closure of groups of tree automorphisms

It is shown that a group defined by forbidding all patterns of size s+1 that do not appear in a given self-similar group of tree automorphisms is the topological closure of a self-similar, countable, regular branch group, branching over its level s stabilizer. As an application, it is shown that there are no infinite, finitely constrained, topologically finitely generated groups of binary tree automorphisms defined by forbidden patterns of size two.

math.GR↗
arXiv2010

Polynomial functions on the units of Z_{2^n}

Polynomial functions on the group of units Q_n of the ring Z_{2^n} are considered. A finite set of reduced polynomials RP_n in Z[x] that induces the polynomial functions on Q_n is determined. Each polynomial function on Q_n is induced by a unique reduced polynomial - the reduction being made using a suitable ideal in Z[x]. The set of reduced polynomials forms a multiplicative 2-group. The obtained results are used to efficiently construct families of exponential cardinality of, so called, huge k-ary quasigroups, which are useful in the design of various types of cryptographic primitives. Along the way we provide a new (and simpler) proof of a result of Rivest characterizing the permutational polynomials on Z_{2^n}.

math.AC↗
arXiv2010

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.

math.GR↗
arXiv2009

On capability of finite abelian groups

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the quotient of some group by its center) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H_i} of subgroups of G with trivial intersection, such that the union generates G and all the quotients G/H_i have the same exponent. The condition that the family of subgroups generates G may be replaced by the condition that the family covers G and the condition that the quotients have the same exponent may be replaced by the condition that the quotients are isomorphic.

math.GR↗
arXiv2008

A pastiche on embeddings into simple groups (following P. E. Schupp)

Let lambda be an infinite cardinal number and let C = {H_i| i in I} be a family of nontrivial groups. Assume that |I|<=lambda, |H_i|<= lambda, for i in I, and at least one member of C achieves the cardinality lambda. We show that there exists a simple group S of cardinality lambda that contains an isomorphic copy of each member of C and, for all H_i, H_j in C with |H_j|=lambda, is generated by the copies of H_i and H_j in S. This generalizes a result of Paul E. Schupp (moreover, our proof follows the same approach based on small cancelation). In the countable case, we partially recover a much deeper embedding result of Alexander Yu. Ol'shanskii.

math.GR↗
arXiv2007

Encouraging the grand coalition in convex cooperative games

A solution function for convex transferable utility games encourages the grand coalition if no player prefers (in a precise sense defined in the text) any coalition to the grand coalition. We show that the Shapley value encourages the grand coalition in all convex games and the tau-value encourages the grand coalitions in convex games up to three (but not more than three) players. Solution functions that encourage the grand coalition in convex games always produce allocations in the core, but the converse is not necessarily true.

math.OC↗
arXiv2007

Schreier spectrum of the Hanoi Towers group on three pegs

Finite dimensional representations of the Hanoi Towers group are used to calculate the spectra of the finite graphs associated to the Hanoi Towers Game on three pegs (the group serves as a renorm group for the game). These graphs are Schreier graphs of the action of the Hanoi Towers group on the levels of the rooted ternary tree. The spectrum of the limiting graph (Schreier graph of the action on the boundary of the tree) is also provided.

math.GR↗
arXiv2007

Frobenius Problem and dead ends in integers

Let a and b be positive, relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In addition, for given integers a and b, we explicitly describe all such elements in Z. Finally, we show that Z has only finitely many dead ends with respect to any finite symmetric generating set. In the appendix we show that every finitely generated group has a generating set with respect to which dead ends exist.

math.NT↗
arXiv2007

The Spectral Problem, Substitutions and Iterated Monodromy

We provide a self-similar measure for the self-similar group $G$ acting faithfully on the binary rooted tree, defined as the iterated monodromy group of the quadratic polynomial $z^2+i$. We also provide an $L$-presentation for $G$ and calculations related to the spectrum of the Markov operator on the Schreier graph of the action of $G$ on the orbit of a point on the boundary of the binary rooted tree.

math.GR↗
arXiv2006

Groups generated by 3-state automata over a 2-letter alphabet, I

An approach to a classification of groups generated by 3-state automata over a 2-letter alphabet and the current progress in this direction are presented. Several results related to the whole class are formulated. In particular, all finite, abelian, and free groups are classified. In addition, we provide detailed information and complete proofs for several groups from the class, with the intention of showing the main methods and techniques used in the classification.

math.GR↗
arXiv2006

Tree morphisms, transducers, and integer sequences

The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms (length and prefix preserving transformations of words). Transducer integer sequences are related to the notion of self-similar groups and semigroups, as well as to the notion of automatic sequences.

math.CO↗
arXiv2006

Hausdorff dimension in a family of self-similar groups

For each prime p and a monic polynomial f, invertible over p, we define a group G_{p,f} of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group G_{2,x^2+x+1}. We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1. We provide a characterization of finitely constrained groups in terms of the branching property, and as a corollary conclude that all defined groups are finitely constrained. In addition, we show that all infinite, finitely constrained groups of p-adic automorphisms have positive and rational Hausdorff dimension and we provide a general formula for Hausdorff dimension of finitely constrained groups. Further ``finiteness'' properties are also discussed (amenability, torsion and intermediate growth).

math.GR↗
arXiv2006

Construction of elements in the closure of Grigorchuk group

We describe constraints that need to be satisfied ``near the top'' of the portraits of the elements in Grigorchuk group. These constraints, if satisfied by the portraits of all sections of some binary tree automorphism, guarantee that this automotphism belongs to the closure of Grigorchuk group in the pro-finite group of binary tree automorphisms. The constraints may be viewed as an effective version of a more conceptual description given earlier by Grigorchuk.

math.GR↗
arXiv2006

Tamari lattices, forests and Thompson monoids

A connection relating Tamari lattices on symmetric groups regarded as lattices under the weak Bruhat order to the positive monoid P of Thompson group F is presented. Tamari congruence classes correspond to classes of equivalent elements in P. The two well known normal forms in P correspond to endpoints of intervals in the weak Bruhat order that determine the Tamari classes. In the monoid P these correspond to lexicographically largest and lexicographically smallest form, while on the level of permutations they correspond to 132-avoiding and 231-avoiding permutations. Forests appear naturally in both contexts as they are used to model both permutations and elements of the Thompson monoid. The connection is then extended to Tamari orders on partitions of ((k-1)n+2)-gons into (k+1)-gons and Thompson monoids P_k, k >1.

math.CO↗