We make a geometric study of the Geometric Rank tensors recently introduced by Kopparty et al. Results include classification with degenerate rank in $C^3\otimes C^3\otimes C^3$, two, and showing that upper bounds on imply lower tensor rank.

The generating rank is determined for several GF(2)-embeddable geometries and it is demonstrated that their generating and embedding ranks are equal. Specifically, we prove that each of the two generalized hexagons of order (2, 2) has generating rank 14, that the central involution geometry of the Hall-Janko sporadic group has generating rank 28, and that the dual polar space DU(6,2) has genera...

A construction is described in [2] by which, given two or more geometries of the same rank n, each equipped with a suitable parallelism giving rise to the same geometry at infinity, we can glue them together along their geometries at infinity, thus obtaininig a new geometry of rank n+ k− 1, k being the number of geometries we glue. In this paper we will examine a special case of that constructi...

By hypotheses (a) and (c), if {i, j} is a 2-element subset of the type set I of Γ, then there are flags F of co-type {i, j}, such that res(F ) is a rank 2 geometry over the type set {i, j}, and all such {i, j}-residues are isomorphic. Hence, such a geometry Γ can be described by a diagram, in which the types (or even the exact isomorphism types) of all rank-2-residues are listed; this is often ...

Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. M. White [Whi02] has proven that the injectivity radius of M is bounded above by some function of rank(π1(M)). Building on a technique that he introduced, we show that if M is a hyperbolic 3-manifold fibering over the circle with fiber Σg and rank(π1(M)) 6= 2g + 1, then the diameter of M ...