International Journal of Algebra and Computation

We explore the relationship between the limit spaces of contracting self-similar groups and self-similar structures. We give the condition on a contracting group such that its limit space admits a self-similar structure, and also the condition such that this self-similar structure is post-critically finite (p.c.f.). We then give necessary and sufficient conditions for a p.c.f. self-similar structure to be the limit space of a contracting self-similar group. When these conditions hold we give a construction of the contracting group. Finally, we illustrate our results with several examples.

At the 2011 Durham Conference "Geometry and Arithmetic of Lattices" M. Kapovich formulated the following

Question. Does there exist an embedding ℤ² * ℤ ↪ SL(3, ℤ)?

The goal of the paper is to prove the following

Main Theorem.If p and m are arbitrary positive integers then there exists an embeddingℤ² * F_m ↪ SL(3, ℤ[1/p]).

In this paper we prove that, if V is a κ-tame pseudovariety which satisfies the pseudoidentity xy^ω+1z = xyz, then the pseudovariety join LSl ∨ V is also κ-tame. Here, LSl denotes the pseudovariety of local semilattices and κ denotes the implicit signature consisting of the multiplication and the (ω – 1)-power. As a consequence, we deduce that LSl ∨ V is decidable. In particular the joins LSl ∨ Ab, LSl ∨ G, LSl ∨ OCR and LSl ∨ CR are decidable.

This paper proves that any monoid presented by a confluent context-free monadic rewriting system is word-hyperbolic. This result then applied to answer a question asked by Duncan and Gilman by exhibiting an example of a word-hyperbolic monoid that does not admit a word-hyperbolic structure with uniqueness (that is, in which the language of representatives maps bijectively onto the monoid).

Let π : Aut(F_n) → Aut(ℤⁿ) be the epimorphism induced by the isomorphism and define . We prove that the subset of consisting of all non-irreducible with irreducible powers and all non-hyperbolic elements is exponentially small.

We present a new solution of the word problem of free algebras in varieties generated by iterated semidirect products of semilattices. As a consequence, we provide asymptotical bounds for free spectra of these varieties. In particular, each finite -trivial (and, dually, each finite -trivial) semigroup has a free spectrum whose logarithm is bounded above by a polynomial function.

We present an algorithm to compute the GIT-fan of algebraic torus actions on affine varieties.

We show that for every quasivariety 𝒦 of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of 𝒦 (the dual of the lattice of sub-quasivarieties of 𝒦) is isomorphic to Con(S, +, 0, 𝒡. As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found.

Part I proved that for every quasivariety 𝒦 of structures (which may have both operations and relations) there is a semilattice S with operators such that the lattice of quasi-equational theories of 𝒦 (the dual of the lattice of sub-quasivarieties of 𝒦) is isomorphic to Con(S, +, 0, 𝒡). It is known that if S is a join semilattice with 0 (and no operators), then there is a quasivariety 𝒬 such that the lattice of theories of 𝒬 is isomorphic to Con(S, +, 0). We prove that if S is a semilattice having both 0 and 1 with a group 𝒢 of operators acting on S, and each operator in 𝒢 fixes both 0 and 1, then there is a quasivariety 𝒲 such that the lattice of theories of 𝒲 is isomorphic to Con(S, +, 0, 𝒢).

We define several "standard" subgroups of the automorphism group Aut(G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut(G). If C is the commutation graph of G, we show how Aut(G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition theorem in the special case where C has no isolated vertices. If C has no vertices of a type we call dominated then we give a semi-direct decomposition of Aut(G) into a subgroup of locally conjugating automorphisms by the subgroup stabilizing a certain lattice of "admissible subsets" of the vertices of C. We then characterize those graphs for which Aut(G) is a product (not necessarily semi-direct) of two such subgroups.

Given tuples a₁, …, a_k and b in Aⁿ for some algebraic structure A, the subpower membership problem asks whether b is in the subalgebra of Aⁿ that is generated by a₁, …, a_k. For A a finite group, there is a folklore algorithm which decides this problem in time polynomial in n and k. We show that the subpower membership problem for any finite Mal'cev algebra is in NP and give a polynomial time algorithm for any finite Mal'cev algebra with finite signature and prime power size that has a nilpotent reduct. In particular, this yields a polynomial algorithm for finite rings, vector spaces, algebras over fields, Lie rings and for nilpotent loops of prime power order.