The isomorphism problem for Coxeter groups

By a recent result obtained by R. Howlett and the author considerable progress has been made towards a complete solution of the isomorphism problem for Coxeter groups. In this paper we give a survey on the isomorphism problem and explain in particular how the result mentioned above reduces it to its `reflection preserving' version. Furthermore we desrcibe recent developments concerning the solution of the latter.Comment: 15 pages, 0 figures, to appear in 'The Coxeter Legacy: Reflections and Projections', Fields Institute Communication

Angle-deformations in Coxeter groups

The isomorphism problem for Coxeter groups has been reduced to its 'reflection preserving version' by B. Howlett and the second author. Thus, in order to solve it, it suffices to determine for a given Coxeter system (W,R) all Coxeter generating sets S of W which are contained in R^W, the set of reflections of (W,R). In this paper, we provide a further reduction: it suffices to determine all Coxeter generating sets S in R^W which are sharp-angled with respect to R.Comment: 23 pages, 6 figures, submitted to AG

Quotients of trees for arithmetic subgroups of PGL₂ over a rational function field

In this note we determine the structure of the quotient of the Bruhat-Tits tree of the locally compact group PGL(2)(F-p) with respect to the natural action of its S-arithmetic subgroup PGL(2)(O-{p}), where F is a rational function field over a finite field and p is a place of F

Codistances of 3-spherical buildings

We show that a 3-spherical building in which each rank 2 residue is connected far away from a chamber, and each rank 3 residue is simply 2-connected far away from a chamber, admits a twinning (i.e., is one half of a twin building) as soon as it admits a codistance, i.e., a twinning with a single chamber.Comment: 35 pages; revised after a referee's comment