International Journal of Algebra and Computation

Answering a question of Dikranjan and Protasov we prove that each infinite group G contains a weakly P-small subset A ⊂ G which is not P-small. This means that for every n there are n pairwise disjoint translation copies of A in G but there is no infinitely many such disjoint copies.

We show that if G ≤ S_A is a permutation group on a finite set A satisfying |A| ≥ 3, then the set of G-closed clones on A that contain all constant operations is finite iff G = S_A, A_A, AGL(1,5) (|A| = 5), PSL(2,5) (|A| = 6), PGL(2,5) (|A| = 6), PGL(2,7) (|A| = 8), PGL(2,8) (|A| = 9), or PΓL(2,8) (|A| = 9).

We consider a very simple Mealy machine (two nontrivial states over a two-symbol alphabet), and derive some properties of the semigroup it generates. It is an infinite, finitely generated semigroup, and we show that the growth function of its balls behaves asymptotically like ℓ^α, for ; that the semigroup satisfies the identity g⁶ = g⁴; and that its lattice of two-sided ideals is a chain.

A classical nilpotency result considers finite p-groups whose proper subgroups all have class bounded by a fixed number n. We consider the analogous property in nilpotent Lie algebras. In particular, we investigate whether this condition puts a bound on the class of the Lie algebra. Some p-group results and proofs carry over directly to the Lie algebra case, some carry over with modified proofs and some fail. For the final of these cases, a certain metabelian Lie algebra is constructed to show a case when the p-groups and Lie algebra cases differ.

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following.

• If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group.

• If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an ℝ-tree is trivial, then H is Hopfian.

• If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian.

• Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier–Wise).

Several transfer results for rational subsets and finitely generated subgroups of HNN-extensions G = 〈 H,t; t^-1 at = φ(a) (a ∈ A) 〉 and amalgamated free products G = H *_A J such that the associated subgroup A is finite. These transfer results allow to transfer decidability properties or structural properties from the subgroup H (resp. the subgroups H and J) to the group G.

Let K be a principal ideal domain, and A_n, with n ≥ 3, be a finitely generated torsion-free abelian group of rank n. Let Ω be a finite subset of KA_n\{0} and U(KA_n) the group of units of KA_n. For a multiplicative monoid P generated by U(KA_n) and Ω, we prove that any generating set for contains infinitely many elements not in . Furthermore, we present a way of constructing elements of not in for n ≥ 3. In the case where K is not a field the aforementioned results hold for n ≥ 2.

Let G be a finitely generated virtually-free group. We consider the Birget–Rhodes expansion of G, which yields an inverse monoid and which is denoted by IM(G) in the following. We show that for a finite idempotent presentation P, the word problem of a quotient monoid IM(G)/P can be solved in linear time on a RAM. The uniform word problem, where G and the presentation P are also part of the input, is EXPTIME-complete. With IM(G)/P we associate a relational structure, which contains for every rational subset L of IM(G)/P a binary relation, consisting of all pairs (x,y) such that y can be obtained from x by right multiplication with an element from L. We prove that the first-order theory of this structure is decidable. This result implies that the emptiness problem for Boolean combinations of rational subsets of IM(G)/P is decidable, which, in turn implies the decidability of the submonoid membership problem of IM(G)/P. These results were known previously for free groups, only. Moreover, we provide a new algorithmic approach for these problems, which seems to be of independent interest even for free groups.

We also show that one cannot expect decidability results in much larger frameworks than virtually-free groups because the subgroup membership problem of a subgroup H in an arbitrary group G can be reduced to a word problem of some IM(G)/P, where P depends only on H. A consequence is that there is a hyperbolic group G and a finite idempotent presentation P such that the word problem is undecidable for some finitely generated submonoid of IM(G)/P. In particular, the word problem of IM(G)/P is undecidable.