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Recently, Leemans and Stokes constructed an infinite family of incidence geometries admitting trialities but no dualities from the groups PSL(2,q) (where q=p3n with p a prime and n>0 a positive integer). Unfortunately, these geometries are not flag transitive. In this paper, we work with the Suzuki groups Sz(q), where q=22e+1 with e a positive integer and 2e+1 is divisible by 3. For any odd integer m dividing q−1, q+2q+1 or q−2q+1 (i.e.: m is the order of some non-involutive element of Sz(q)), we construct geometries of type (m,m,m) that admit trialities but no dualities. We then prove that they are flag transitive when m=5, no matter the value of q. These geometries form the first infinite family of incidence geometries of rank 3 that are flag transitive and have trialities but no dualities. They are constructed using chamber systems and the trialities come from field automorphisms. These same geometries can also be considered as regular hypermaps with automorphism group Sz(q).

We investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.

We introduce the notion of moving absolute geometry of a geometry with triality and show that, in the classical case where the triality is of type (Iσ) and the absolute geometry is a generalized hexagon, the moving 5 3 6 absolute geometry also gives interesting flag-transitive geometries with Buekenhout diagram for the groups G2(k) and 3D4(k), for any prime power k ≥ 2. We also classify the absolute geometries for geometries with trialities but no dualities coming from maps of Class III with automorphism group L2(q3), where q ≥ 2 is prime power. We then investigate the moving absolute geometries for these geometries, illustrating their interest in this case.

We present results on rigidity of structures of rigid rods connected in joints: rod configurations. The underlying combinatorial structure of a rod configuration is an incidence structure. Our aim is to find simple ways of determining which rod configurations admit non-trivial motions, using the underlying incidence structure.

Rigidity of graphs in the plane is well understood. Indeed, there is a polynomial time algorithm for deciding whether most realisations of a graph are rigid. One of the results presented here equates rigidity of sufficiently generic rod configurations to rigidity of a related graph. As a consequence, itis possible to determine the rigidity of rod configurations using the previously mentioned polynomial time algorithm. We use this to show that all v3-configurations on up to 15 points and all triangle-free v3-configurations on up to 20 points are rigid in regular position, if such a realisation exists. We also conjecture that the smallest v3-configuration that is flexible in regular position is a previously known 28₃-configuration. 

In this article we explore the rigidity of realizations of incidence geometries consisting of points and rigid rods: rod configurations. We survey previous results on the rigidity of structures that are related to rod configurations, discuss how to find realizations of incidence geometries as rod configurations, and how this relates to the 2-plane matroid. We also derive further sufficient conditions for the minimal rigidity of k-uniform rod configurations and give an example of an infinite family of minimally rigid 3-uniform rod configurations failing the same conditions. Finally, we construct v3-configurations that are flexible in the plane, and show that there are flexible v3-configurations for all sufficiently large values of v.

Triality is a classical notion in geometry that arose in the context of the Lie groups of type D4. Another notion of triality, Wilson triality, appears in the context of reflexible maps. We build a bridge between these two notions, showing how to construct an incidence geometry with a triality from a map that admits a Wilson triality. We also extend a result by Jones and Poulton, showing that for every prime power q, the group L2 (q3) has maps that admit Wilson trialities but no dualities.

New results on pentagonal geometries PENT(k, r) with block sizes k = 3 or k = 4 are given. In particular we completely determine the existence spectra for PENT(3, r) systems with the maximum number of opposite line pairs as well as those without any opposite line pairs. A wide-ranging result about PENT(3, r) with any number of opposite line pairs is proved. We also determine the existence spectrum of PENT(4, r) systems with eleven possible exceptions.

We present an inventory and the enumeration of all non-isomorphic rigidity circuits on up to 10 vertices, as well as all non-isomorphic Assur graphs on up to 11 vertices. Assur graphs and Baranov trusses are closely related. We clarify the relation between Baranov trusses and (2, 3)-tight graphs on the one hand, and between Assur groups and Assur graphs on the other hand.