Some free-by-cyclic groups

We exhibit free-by-cyclic groups containing non-free locally-free subgroups, including some word hyperbolic examples. We also show that these groups are not subgroup separable. We use Bestvina-Brady Morse theory in our arguments

Property A and CAT(0) cube complexes

Property A is a non-equivariant analogue of amenability defined for metric spaces. Euclidean spaces and trees are examples of spaces with Property A. Simultaneously generalising these facts, we show that finite-dimensional CAT(0) cube complexes have Property A. We do not assume that the complex is locally finite. We also prove that given a discrete group acting properly on a finite-dimensional CAT(0) cube complex the stabilisers of vertices at infinity are amenable

Whole-genome sequencing reveals host factors underlying critical COVID-19

Critical COVID-19 is caused by immune-mediated inflammatory lung injury. Host genetic variation influences the development of illness requiring critical care1 or hospitalization2,3,4 after infection with SARS-CoV-2. The GenOMICC (Genetics of Mortality in Critical Care) study enables the comparison of genomes from individuals who are critically ill with those of population controls to find underlying disease mechanisms. Here we use whole-genome sequencing in 7,491 critically ill individuals compared with 48,400 controls to discover and replicate 23 independent variants that significantly predispose to critical COVID-19. We identify 16 new independent associations, including variants within genes that are involved in interferon signalling (IL10RB and PLSCR1), leucocyte differentiation (BCL11A) and blood-type antigen secretor status (FUT2). Using transcriptome-wide association and colocalization to infer the effect of gene expression on disease severity, we find evidence that implicates multiple genes—including reduced expression of a membrane flippase (ATP11A), and increased expression of a mucin (MUC1)—in critical disease. Mendelian randomization provides evidence in support of causal roles for myeloid cell adhesion molecules (SELE, ICAM5 and CD209) and the coagulation factor F8, all of which are potentially druggable targets. Our results are broadly consistent with a multi-component model of COVID-19 pathophysiology, in which at least two distinct mechanisms can predispose to life-threatening disease: failure to control viral replication; or an enhanced tendency towards pulmonary inflammation and intravascular coagulation. We show that comparison between cases of critical illness and population controls is highly efficient for the detection of therapeutically relevant mechanisms of disease

Engulfing in word-hyperbolic groups

We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups

The singularity obstruction for group splittings

We study an obstruction to splitting a finitely generated group G as an amalgamated 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

Finding splittings of groups and three-manifolds

The class of almost finitely presented groups which admit a finite index subgroup which is an HNN extension is characterised using Scott's end invariant for a pair of groups. As in Scott's original paper on the subject the profinite topology plays a role

The subgroup separability of some amalgamated free products

SIGLEAvailable from British Library Document Supply Centre- DSC:DX95180 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

K-theory for subspaces of groups

We show that a subspace of a group carries a natural partial translation structure which gives rise to a C*-algebra similar to that of a reduced C*-algebra of a group. We investigate the question: Does the inclusion of a subspace into an ambient group induce a homomorphism of C*-algebras? We prove that the answer is affirmative for subspaces of groups with non-coarsely dense complement. We show that under this condition there exists an exact sequence of C*-algebras which is analogous to the Pimsner-Voiculescu extension

The geometry of cube complexes and the complexity of their fundamental groups

We investigate the geometry of geodesics in CAT(0) cube complexes. A group which acts cocompactly and properly discontinuously on such a complex is shown to have a biautomatic structure. There is a family of natural subgroups each of which is shown to be rational