Two-ended quasi-transitive graphs

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation [Formula: see text] or an HNN-extension [Formula: see text], where [Formula: see text] is a finite group and [Formula: see text] and [Formula: see text]. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

On the lower central series of some virtual knot groups

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual knot groups as semi direct product and free product with amalgamation. In particular, we prove that the groups of some virtual knots are extensions of finitely generated free groups by infinite cyclic groups.

Completely metrisable groups acting on trees

We consider actions of completely metrisable groups on simplicial trees in the context of the Bass–Serre theory. Our main result characterises continuity of the amplitude function corresponding to a given action. Under fairly mild conditions on a completely metrisable groupG, namely, that the set of elements generating a non-discrete or finite subgroup is somewhere dense, we show that in any decomposition as a free product with amalgamation,G=A*cB, the amalgamated groupsA,BandCare open inG.

BICYCLIC SUBSEMIGROUPS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max {|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the [Formula: see text]-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the [Formula: see text]-classes to be finite.

PROPERTY (FA) AND LATTICES IN SU(2,1)

In this paper we consider Property (FA) for lattices in SU(2,1). First, we prove that [Formula: see text] has Property (FA). We then prove that the arithmetic lattices in SU(2,1) of second type arising from congruence subgroups studied by Rapoport–Zink and Rogawski cannot split as a nontrivial free product with amalgamation; one such example is Mumford's fake projective plane. In fact, we prove that the fundamental group of any fake projective plane has Property (FA).

AUTOMATIC STRUCTURES FOR FUNDAMENTAL GROUPOIDS OF GRAPHS OF GROUPS

We give sufficient conditions for the fundamental groupoid of a graph of groups to be automatic (resp. asynchronously automatic). These conditions are similar to those in [1] for a free product with amalgamation to be automatic (resp. asynchronously automatic).

On Certain Finitely Generated Subgroups of Groups Which Split

Define a group G to be in the class 𝒮 if for any finitely generated subgroup K of G having the property that there is a positive integer n such that gn ∈ K for all g ∈ G, K has finite index in G. We show that a free product with amalgamation A *CB and an HNN group A *C belong to 𝒮, if C is in 𝒮 and every subgroup of C is finitely generated.

ON THE WORD PROBLEM FOR TENSOR PRODUCTS AND AMALGAMS OF MONOIDS

We prove that the word problem for the free product with amalgamation S *UT of monoids can be undecidable, even when S and T are finitely presented monoids with word problems that are decidable in linear time, the factorization problems for U in each of S and T, as well as other problems, are decidable in polynomial time, and U is a free finitely generated unitary submonoid of both S and T. This is proved by showing that the equality problem for the tensor product S ⊗UT is undecidable and using known connections between tensor products and amalgams. We obtain similar results for semigroups, and by passing to semigroup rings, we obtain similar results for rings as well. The proof shows how to simulate an arbitrary Turing machine as a communicating pair of two deterministic pushdown automata and is of independent interest. A similar idea is used in a paper by E. Bach to show undecidability of the tensor equality problem for modules over commutative rings.

Free Products with Amalgamation and p-Adic Lie Groups

Using the theory of p-adic Lie groups we give conditions for a finitely generated group to admit a splitting as a non-trivial free product with amalgamation. This can be viewed as an extension of a theorem of Bass.