Abstract. We introduce a method to explicitly determine the Farrell-Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL2(O) for O the ring of integers in an imaginary quadratic number field, and to their finite index subgroups. We show that the Farrell-Tate cohomology of the Bianchi groups is completely determined by the numbers of conjugacy classes of finite subgroups. In fact, our access to Farrell-Tate cohomology allows us to detach the information about it from geometric models for the Bianchi groups and to express it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups have been determined in a thesis of Krämer, in terms of elementary number-theoretic information on O. An evaluation of these formulae for a large number of Bianchi groups is provided numerically in the appendix.Our new insights about their homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups.

Abstract. The Bianchi groups are the groups (P)SL 2 over a ring of integers in an imaginary quadratic number field. We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups.Furthermore, this correspondence facilitates the computation of the equivariant K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is satisfied by the Bianchi groups, we obtain the K-theory of their reduced C * -algebras in terms of isomorphic images of their equivariant Khomology.

The human gut microbiome includes beneficial, commensal and pathogenic bacteria that possess antimicrobial resistance (AMR) genes and exchange these predominantly through conjugative plasmids. Escherichia coli is a significant component of the gastrointestinal microbiome and is typically non-pathogenic in this niche. In contrast, extra-intestinal pathogenic E. coli (ExPEC) including ST131 may occupy other environments like the urinary tract or bloodstream where they express genes enabling AMR and host cell adhesion like type 1 fimbriae. The extent to which commensal E. coli and uropathogenic ExPEC ST131 share AMR genes remains understudied at a genomic level, and we examined this here using a preterm infant resistome. We found that individual ST131 had small differences in AMR gene content relative to a larger shared resistome. Comparisons with a range of plasmids common in ST131 showed that AMR gene composition was driven by conjugation, recombination and mobile genetic elements. Plasmid pEK499 had extended regions in most ST131 Clade C isolates, and it had evidence of a co-evolutionary signal based on protein-level interactions with chromosomal gene products, as did pEK204 that had a type IV fimbrial pil operon. ST131 possessed extensive diversity of selective type 1, type IV, P and F17-like fimbriae genes that was highest in subclade C2. The structure and composition of AMR genes, plasmids and fimbriae vary widely in ST131 Clade C and this may mediate pathogenicity and infection outcomes.