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

Finite complete rewriting systems for regular semigroups

It is proved that, given a (von Neumann) regular semigroup with finitely many left and right ideals, if every maximal subgroup is presentable by a finite complete rewriting system, then so is the semigroup. To achieve this, the following two results are proved: the property of being defined by a finite complete rewriting system is preserved when taking an ideal extension by a semigroup defined by a finite complete rewriting system; a completely 0-simple semigroup with finitely many left and right ideals admits a presentation by a finite complete rewriting system provided all of its maximal subgroups do.Comment: 11 page

(1RS,2SR,7RS,8RS)-N-Benzoyltricyclo[6.2.2.0²,⁷]dodeca-9,11-diene-1,10-dicarboximide

The title 1,4-photoadduct, C₂₁H₁₉NO₃, was formed on irradiation of N-benzoylphthalimide in dichloromethane containing cyclohexene. The bond lengths and angles are generally within the normal ranges. A notable feature of the molecule is the presence within it of four contiguous chiral centres

4',5',6',7'-Tetrachlorospiro[cyclohex-2-ene-1,2'-indan]-1',3'-dione

The title compound, C₁₄H₈Cl₄O₂, has been isolated following irradiation of a dichloromethane solution of N-acetyltetrachlorophthalimide and cyclohexene. The structure refinement is slightly compromised by the disorder over two positions of equal occupancy of a methylene groupβ to the spiro C atom

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

Benzylammonium 2,4-bis(dicyanomethylene)-2,3-dihydroisoindolide

The cation and anion of the title salt, C⁷H₁₀N⁺.C₁₄H₄N₅-, are both bisected by a crystallographic mirror plane. Extensive hydrogen bonding, with the R₆⁶(28) graph-set motif, connects the ions into layers

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