International Journal of Algebra and Computation

An infinite graph Γ is minor excluded if there is a finite graph that is not a minor of Γ. We prove that minor excluded graphs have finite Assouad–Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators, like ℤ³; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.

We investigate the function d_A(n), which gives the size of a least size generating set for Aⁿ, in the case where A has a cube term. We show that if A has a k-cube term and A^k is finitely generated, then d_A(n) ∈ O(log(n)) if A is perfect and d_A(n) ∈ O(n) if A is imperfect. When A is finite, then one may replace "Big O" with "Big Theta" in these estimates.

A monoid M is called surjunctive if every injective cellular automata with finite alphabet over M is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive.

To our knowledge at the time of writing, the maximum Waring rank for the set of all ternary forms of degree d (with coefficients in an algebraically closed field of characteristic zero) is known only for d ≤ 4, and the best upper bound that is known for d = 5 is 12. In this work, we lower the upper bound to 10 (and there are evidences that it is likely sharp).

The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa–Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych. In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be significant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.

In this paper, we describe the automorphism groups of the endomorphism semigroups of free Burnside groups B(m, n) for odd exponents n ≥ 1003. We prove, that the groups Aut(B(m, n)) and Aut(End(B(m, n))) are canonically isomorphic. In particular, if the groups Aut(End(B(m, n))) and Aut(End(B(k, n))) are isomorphic, then m = k.

For each d ∈ ℕ, we construct a 3-generated group H_d, which is a subdirect product of free groups, such that the cohomological dimension of H_d is d. Given a group F and a normal subgroup N ⊳ F we prove that any right angled Artin group containing the special HNN-extension of F with respect to N must also contain F/N. We apply this to construct, for every d ∈ ℕ, a 4-generated group G_d, embeddable into a right angled Artin group, such that the cohomological dimension of G_d is 2 but the cohomological dimension of any right angled Artin group, containing G_d, is at least d. These examples are used to show the non-existence of certain "universal" right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular, we show that for every n ∈ ℕ there exists δ(n) ∈ ℕ such that any n-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the δ(n)-th direct power of the free group of rank 2. As another corollary we derive that any n-generated finitely presented residually free group embeds into the direct product of at most δ(n) limit groups.