This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.

We classify all mutation-finite quivers with real weights. We show that every finite mutation class not originating from an integer skew-symmetrisable matrix has a geometric realisation by reflections. We also explore the structure of acyclic representatives in finite mutation classes and their relations to acute-angled simplicial domains in the corresponding reflection groups.

We call a packing of hyperspheres in n dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all; they fill the n-dimensional Euclidean space; and every sphere in the packing is a member of a cluster of $n+2$ mutually tangent spheres (and a few more properties described herein). In this paper, we describe an Apollonian packing in eight dimensions that naturally arises from the study of generic nodal Enriques surfaces. The $E_7$, $E_8$ and Reye lattices play roles. We use the packing to generate an Apollonian packing in nine dimensions, and a cross section in seven dimensions that is weakly Apollonian. Maxwell described all three packings but seemed unaware that they are Apollonian. The packings in seven and eight dimensions are different than those found in an earlier paper. In passing, we give a sufficient condition for a Coxeter graph to generate mutually tangent spheres and use this to identify an Apollonian sphere packing in three dimensions that is not the Soddy sphere packing.

The Gehring–Martin–Tan inequality for two-generator subgroups of $\text{PSL}(2,\mathbb{C})$ is one of the best known discreteness conditions. A Kleinian group $G$ is called a Gehring–Martin–Tan group if the equality holds for the group $G$ . We give a method for constructing Gehring–Martin–Tan groups with a generator of order four and present some examples. These groups arise as groups of finite-volume hyperbolic 3-orbifolds.

Let R be a not necessarily commutative local ring, M a free R-module, and π ∈ GL(M) such that B(π) = im(π –1)is a subspace of M. Then π = σ1…σtρ, where σi are simple mappings of given types, ρ is a simple mapping, B(sgr;i) and B(ρ) are subspaces and t ≤ dim B(π).