Cyclically presented groups Gn(w) in which w involves at most three generators are studied. We classify when such groups have infinite abelianizations and apply this result to the groups Gn(x1x1+kx 1+`). As a further corollary we give sufficient conditions for the natural HNN extension of Gn(w) to be a high-dimensional knot group. By continuing Cavicchioli, Repovš, and Spaggiari’s investigation...

We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in ...

A generalized triangle group is a group that can be presented in the form G = x, y | x p = y q = w(x, y) r = 1 where p, q, r ≥ 2 and w(x, y) is a cyclically reduced word of length at least 2 in the free product Z p * Z q = x, y | x p = y q = 1. Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibl...

We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andrásfai graph, Doob graph, cocktail party graph, crown graph, prism graph, Möbius ladder. The proofs operate by finding the abelianisation of a cyclically presented...

In a focus+context visualization, one often finds it difficult to overlay focus information on top of a dense visual context. This work is motivated by a need for visualizing search results (i.e., focus) in relation to a treemap representation of a large file system (i.e., context). We thus consider that the focus consists of a collection of visual objects discretely-distributed over a backgrou...

We observe that the group of all lifts elements Thompson’s T T to real line is finitely presented and contains additive alttext="double...

In 1979, Rourke proposed to extend the set of cyclically reduced defining words of a group presentation P by using operations of cyclic permutation, inversion and taking double products. He proved that iterations of these operations yield all cyclically reduced words of the normal closure of defining words of P if the group, defined by the presentation P, is trivial. We generalize this result b...

In this paper, we found the cyclically presented groups obtained from the word w generated with some Dunwoody parameters. 2000 Mathematics Subject Classification: 57M05, 20F38