In this paper, by using the Gröbner-Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group.

The history, definition and principal properties of automatic groups and their generalisations to subgroups and cosets are reviewed briefly, mainly from a computational perspective. A result about the asynchronous automaticity of an HNN extension is then proved and applied to an example that was proposed by Mark Sapir. AMS Classification 20F32; 20F05

This study addresses the challenge of assessing carcinogenic potential hazardous chemical mixtures, such as per- and polyfluorinated substances (PFASs), which are known to contribute significantly cancer development. Here, we propose a novel framework called HNNMixCancer that utilizes hybrid neural network (HNN) integrated into machine-learning framework. incorporates mathematical model simulat...

Mononitrobiuret (MNB) and 1,5-Dinitrobiuret (DNB) are, tetrazole-free, nitrogen-rich, energetic compounds. For the first time, the thermal decomposition mechanisms of MNB and DNB have been investigated by advanced ab initio kinetics theories. In particular, the intramolecular interactions of amine H-atom with electronegative nitro O-atom and carbonyl O-atom have been analyzed for MNB, DNB, and ...

In this article we consider transient random walks on HNN extensions of finitely generated groups. We prove that the rate escape w.r.t. some generalised word length exists. Moreover, a central limit theorem with respect to is derived. Finally, show escape, which can be regarded as function in many parameters describe walk, behaves real-analytic terms probability measures constant support.

We study the property of tame combability for groups. We show that quasi-isometries preserve this property. We prove that an amalgamation, A ∗C B, where C is finitely generated, is tame combable iff both A and B are. An analogous result is obtained for HNN extensions. And we show that all one-relator groups are tame combable.

We study an obstruction to splitting a finitely generated group G as an amalagamated free product or HNN extension over a given subgroup H and show that when the obstruction is “small” G splits over a related subgroup. Applications are given which generalise decomposition theorems from low dimensional topology.

We show that a group presented by a labelled oriented tree presentation in which the tree has diameter at most three is an HNN extension of a finitely presented group. From results of Silver, it then follows that the corresponding higher dimensional ribbon knots admit minimal Seifert manifolds. AMS Classification 57Q45; 20E06, 20F05, 57M05

We show that if the fundamental group of the complement of a rationally homologically fibered knot in a rational homology 3sphere is bi-orderable, then its Alexander polynomial has at least one positive real root. Our argument can be applied for a finitely generated group which is an HNN extension with certain properties.

We prove that the natural HNN-extensions of the fractional Fibonacci groups are the fundamental groups of high-dimensional knot complements. We also give some characterization and interpretation of these knots. In particular we show that some of them are 2-knots.