In 2006 we completed the proof of a five-part conjecture which was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2-generator, 2-relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theore...

We obtain a criterion for quasiconvexity of a subgroup of an amalgamated free product of two word hyperbolic groups along a virtually cyclic subgroup. The result provides a method of constructing new word hyperbolic group in class (Q), that is such that all their finitely generated subgroups are quasiconvex. It is known that free groups, hyperbolic surface groups and most 3-dimensional Kleinian...

We study the quotient of a free product of groups by the normal closure of a word that is contained in a certain type of 2-generator subgroup, and hence can be expressed in the form of a pushout involving a generalized triangle group. Using the theory of generalized triangle groups, we extend the range of conditions under which a number of results for one-relator products of groups, such as the...

It is a well-known fact that every group G has a presentation of the form G = F/R, where F is a free group and R the kernel of the natural epimorphism from F onto G. Driven by the desire to obtain a similar presentation of the group of automorphisms Aut(G), we can consider the subgroup Stab(R) ⊆ Aut(F ) of those automorphisms of F that stabilize R, and try to figure out if the natural homomorph...

Garside groups are a natural lattice-theoretic generalisation of the braid and spherical type Artin–Tits groups. Here we show that class is closed under some free products with cyclic amalgamated subgroups. We deduce every tree product infinite group. Moreover, study those HNN extensions as well. Using theorem Pietrowski, conclude this paper by stating non-cyclic one-relator group if only its c...

We show that for any finite group G and for any d there exists a word w ∈ Fd such that a d-tuple in G satisfies w if and only if it generates a solvable subgroup. In particular, if G itself is not solvable, then it cannot be obtained as a quotient of the one relator group Fd/ 〈w〉. As a corollary, the probability that a word is satisfied in a fixed nonsolvable group can be made arbitrarily small...