On geometric aspects of diffuse groups

Bowditch introduced the notion of diffuse groups as a geometric variation of the unique product property. We elaborate on various examples and non-examples, keeping the geometric point of view from Bowditch's paper. In particular, we discuss fundamental groups of flat and hyperbolic manifolds. The appendix settles an open question by providing an example of a group which is diffuse but not left-orderable.Comment: 37 pages, main text by Kionke and Raimbault, appendix by Dunfield. v2 : updated ancillary files and added url to the reference

Coxeter polytopes and Benjamini--Schramm convergence

We observe that a large part of the volume of a hyperbolic polyhedron is taken by a tubular neighbourhood of its boundary, and use this to give a new proof for the finiteness of arithmetic maximal reflection groups following a recent work with M. Fraczyk and S. Hurtado. We also study in more depth the case of polygons in the hyperbolic plane

On distinct finite covers of 3-manifolds

Every closed orientable surface S has the following property: any two connected covers of S of the same degree are homeomorphic (as spaces). In this, paper we give a complete classification of compact 3-manifolds with empty or toroidal boundary which have the above property. We also discuss related group-theoretic questions.Comment: 29 pages. V3: Implements suggestions from a referee report. This version has been accepted for publication by IUM

Reply to: Comment on `Long-range electrostatic interactions between like-charged colloids: steric and confinement effects'

In his Comment (cond-mat/0104060) to [Phys. Rev. E 60, 6530 (1999)], Mateescu shows that while the effective interactions remain repulsive when the specific size of the micro-ions is taken into account via a Modified Poisson-Boltzmann equation, a similar conclusion cannot be reached for the situation of complete lateral confinement. This point is correct but has already been considered in a more general study [Phys. Rev. E 62, R1465 (2000), where repulsion is generically obtained]; moreover, we argue that it illustrates the irrelevancy of the notion of pair potential in completely confined configurations, as shown on a simple example