On one-relator products induced by generalized triangle groups II

Abstract

In this paper we study a group G which is the quotient of a free product of groups by the normal closure of a word that is contained in a in a subgroup which has the form of a generalised triangle group. We use known properties of generalised triangle groups, together with detailed analyses of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The hypotheses are weaker than those in an earlier article of Howie and Shwartz, yielding generalisations in two directions of the results in that article.

Structure of words with short 2-length in a free product of groups

“…We also mention that the case when m = 2 is largely open. For partial results in this case see [5,1,2]. The aim of this paper is to extend the result in [4] by allowing to an extent letters of order 2 in R. In [4] it was shown that R m satisfies the small cancellation condition C(2m) when D 2 (S R ) = 0, which is essentially an observation in [11].…”

“…In Section section 3 we prove a number of lemmas about word combinatorics and pictures. These lemmas are then used to deduce Theorems [1][2]. In Section section 4 we deduce a number of applications of our results.…”

“…Below, we recall only basic concepts on pictures over a one-relator product. For more details, the reader can see [6,7,8,4,2].…”

“…There is an already developed theory for one-relator products of type E(p, q, m) (see [9,1,2]). Hence by Theorem 1, we can apply this theory in our setting.…”

Structure of words with short 2-length in a free product of groups

“…We also mention that the case when m = 2 is largely open. For partial results in this case see [9,10,11]. The aim of this paper is to extend the result in [7] by allowing letters of order 2 in R. Also we require results about pictures over G, in particular the fact that R m satisfies the small cancellation condition C(2m) when R has a certain form.…”