Combinatorial rigidity and symmetry

Louis Theran (Freie Universität Berlin)

Abstract :

Bar-joint frameworks are structures made of fixed-length bars connected by universal joints with full rotational freedom. The allowed motions preserve the length and connectivity of the bars, and a framework is rigid if all the allowed motions extend to rigid body motions. For finite frameworks with generic geometry, rigidity is known to depend only on the graph that has as its edges the bars, and, in dimension 2, the generically rigid graphs are known exactly. In recent years, the question of extending this kind of combinatorial theory to infinite frameworks or finite frameworks with special geometry such as symmetry has received a lot of attention, motivated, in part, by applications in crystallography.

I will review some combinatorial characterizations of rigidity for 2-dimensional "crystallographic" bar-joint frameworks which are infinite, symmetric with respect to a crystallographic group $\Gamma$, and constrained to move in a way that preserves $\Gamma$-symmetry. Then I will discuss a new, finite algebraic characterization of so-called "ultrarigid" frameworks that are symmetric with respect to a full-rank translation lattice $\Lambda$, but allowed to move symmetrically with respect to any finite-index sub-lattice $\Lamba' < \Lambda$.

This is joint work with Justin Malestein.