Subgroups of direct products of two limit groups

If S is a subgroup of a direct product of two limit groups, and S is of type FP(2) over the rationals, then S has a subgroup of finite index that is a direct product of at most two limit groups.Comment: 18 pages, no figure

Normalisers in Limit Groups

Let \G be a limit group, S\subset\G a subgroup, and $N$ the normaliser of $S$. If $H_1(S,\mathbb Q)$ has finite \Q-dimension, then $S$ is finitely generated and either $N/S$ is finite or $N$ is abelian. This result has applications to the study of subdirect products of limit groups.Comment: 10 pages, no figure

Subgroups of direct products of elementarily free groups

We exploit Zlil Sela's description of the structure of groups having the same elementary theory as free groups: they and their finitely generated subgroups form a prescribed subclass E of the hyperbolic limit groups. We prove that if $G_1,...,G_n$ are in E then a subgroup $\Gamma\subset G_1\times...\times G_n$ is of type \FP_n if and only if $\Gamma$ is itself, up to finite index, the direct product of at most $n$ groups from $\mathcal E$. This answers a question of Sela.Comment: 19 pages, no figure

Derivatisation of Polyphenols

Polyphenols, such as tannins, offer potential as a bio-derived chemical feedstock. Their present utilisation is limited mainly to leather tanning and wood panel adhesives. However, appropriate derivatisation may alter both the chemical and physical properties and thereby allow further utilisation of polyphenols. Derivatisation of polyphenols was achieved by esterification and etherification of the phenol groups. Esterification was achieved by alcoholysis of acid chlorides and transesterification with vinyl esters, while etherification was achieved by the ring opening of propylene oxide. The polyphenols used were resorcinol, catechin, Pinus radiata bark tannin, and Schinopsis lorentzii tannin. The products were characterised using a range of techniques including NMR (1H, 13C and 2D NMR in both the solution and solid state), ESI-MS, GPC, DSC, TGA, and rheology. The preparation of polyphenolic esters by alcoholysis provided model compounds to establish the key chemical, spectroscopic, and physical features. A range of simple polyphenol esters such as resorcinol dilaurate and catechin pentalaurate were prepared using lauroyl chloride. Furthermore, tannin lauroyl esters were prepared with varying degrees of substitution. A transesterification method was developed for the preparation of polyphenol esters. Ester interchange occurred effectively in the presence of base catalyst in aqueous solution or dimethyl sulfoxide with short or long chain vinyl esters. This included the first report of the base-catalysed transesterification of flavonoids by vinyl esters to give products such as catechin mono- and di-laurate. Transesterification occurred preferentially at the B-ring as shown by NMR spectroscopy. Subsequently, this transesterification procedure was used to prepare tannin esters. The chemical and physical properties of polyphenol esters were assessed using thermal, antioxidant, and UV/VIS light absorption analysis. Thermal analysis indicated melt/flow properties for some of the polyphenol esters. In some cases, the thermal stability was also shown to increase upon esterification. The antioxidant activity was shown to decrease upon transesterification of pine bark tannin with vinyl laurate, while the UV/VIS absorption was shown to increase. These properties may lend the products towards applications as polymer additives or pharmaceuticals. Polyphenol ethers were prepared by the Williamson ether synthesis and ring opening of propylene oxide. However, the Williamson ether synthesis, a common route to prepare ethers, proved unsuitable for flavonoids. Catechin and tannin hydroxypropyl ether derivatives of varying substitution were prepared by the ring-opening of propylene oxide in the presence of triethylamine. Upon hydroxypropylation the thermal properties of the polyphenol were altered. For example, catechin hydroxypropyl ethers showed a glass transition, which was dependent upon the molar substitution, while rheology showed melt behaviour for several of the tannin hydroxypropyl ethers

Subgroups of direct products of limit groups

If $G_1,...,G_n$ are limit groups and $S\subset G_1\times...\times G_n$ is of type \FP_n(\mathbb Q) then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This settles a question of Sela.Comment: 20 pages, no figures. Final version. Accepted by the Annals of Mathematic

On the finite presentation of subdirect products and the nature of residually free groups

We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $\Gamma_1,...,\Gamma_n$ are finitely presented and $S<\Gamma_1\times...\times\Gamma_n$ projects to a subgroup of finite index in each $\Gamma_i\times\Gamma_j$, then $S$ is finitely presentable, indeed there is an algorithm that will construct a finite presentation for $S$. We use the VSP criterion to characterise the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\rm{FP}}_\infty$ nor of Stallings-Bieri typeComment: 44 pages. To appear in American Journal of Mathematics. This is a substantial rewrite of our previous Arxiv article 0809.3704, taking into account subsequent developments, advice of colleagues and referee's comment

Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups

We study a class of Labelled Oriented Graph (LOG) group where the underlying graph is a tadpole graph. We show that such a group is the natural HNN extension of a cyclically presented group and investigate the relationship between the LOG group and the cyclically presented group. We relate the second homotopy groups of their presentations and show that hyperbolicity of the cyclically presented group implies solvability of the conjugacy problem for the LOG group. In the case where the label on the tail of the LOG spells a positive word in the vertices in the circuit we show that the LOGs and groups coincide with those considered by Szczepa�nski and Vesnin. We obtain new presentations for these cyclically presented groups and show that the groups of Fibonacci type introduced by Johnson and Mawdesley are of this form. These groups generalize the Fibonacci groups and the Sieradski groups and have been studied by various authors. We continue these investigations, using small cancellation and curvature methods to obtain results on hyperbolicity, automaticity, SQ-universality, and solvability of decision problems

Multi-resolution isotropic strain limiting

In this paper we describe a fast strain-limiting method that allows stiff, incompliant materials to be simulated efficiently. Unlike prior approaches, which act on springs or individual strain components, this method acts on the strain tensors in a coordinate-invariant fashion allowing isotropic behavior. Our method applies to both two-and three-dimensional strains, and only requires computing the singular value decomposition of the deformation gradient, either a small 2x2 or 3x3 matrix, for each element. We demonstrate its use with triangular and tetrahedral linear-basis elements. For triangulated surfaces in three-dimensional space, we also describe a complementary edge-angle-limiting method to limit out-of-plane bending. All of the limits are enforced through an iterative, non-linear, Gauss-Seidel-like constraint procedure. To accelerate convergence, we propose a novel multi-resolution algorithm that enforces fitted limits at each level of a non-conforming hierarchy. Compared with other constraint-based techniques, our isotropic multi-resolution strain-limiting method is straightforward to implement, efficient to use, and applicable to a wide range of shell and solid materials. © 2010 ACM