Computing S-unit groups of orders

Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of [Formula: see text]-unit groups of orders. The algorithm is then used for some investigations concerning the congruence subgroup property.

The profinite completion of multi-EGS groups

The class of multi-EGS groups is a generalisation of the well-known Grigorchuk–Gupta–Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally, our results show that branch multi-EGS groups are just infinite.

Profinite invariants of arithmetic groups

We prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results for $\ell^2$ -torsion as well as a strong profiniteness statement for Novikov–Shubin invariants.

Multi-GGS Groups have the Congruence Subgroup Property

We generalize the result about the congruence subgroup property for GGS groups in [3] to the family of multi-GGS groups; that is, all multi-GGS groups except the one defined by the constant vector have the congruence subgroup property. New arguments are provided to produce this more general proof.

On the lattice of subgroups of the lamplighter group

The techniques of modules and actions of groups on rooted trees are applied to study the subgroup structure and the lattice subgroup of lamplighter type groups of the form ℒn,p = (ℤ/pℤ)n ≀ ℤ for n ≥ 1 and p prime. We completely characterize scale invariant structures on ℒ1,2. We determine all points on the boundary of binary tree (on which ℒ1,p naturally acts in a self-similar manner) with trivial stabilizer. We prove the congruence subgroup property (CSP) and as a consequence show that the profinite completion [Formula: see text] of ℒ1,p is a self-similar group generated by finite automaton. We also describe the structure of portraits of elements of ℒ1,p and [Formula: see text] and show that ℒ1,p is not a sofic tree shift group in the terminology of [T. Ceccherini-Silberstein, M. Coornaert, F. Fiorenza and Z. Sunic, Cellular automata between sofic tree shifts, Theor. Comput. Sci.506 (2013) 79–101; A. Penland and Z. Sunic, Sofic tree shifts and self-similar groups, preprint].

A PROBABILISTIC ZETA FUNCTION FOR ARITHMETIC GROUPS

A profinite group G is positively finitely generated (PFG) if for some k, the probability P(G,k) that k random elements generate G is positive. It was conjectured that if G is PFG, then the function P(G,k) can be interpolated to an analytic function defined in some right half-plane. Here that conjecture is formulated more precisely, and verified for (the profinite completion of) arithmetic groups satisfying the congruence subgroup property.