The generalized Markoff equation gives rise to a dynamical system via the Markoff group action on its solution set. Over finite fields, the action produces graphs that are conjectured by Bourgain, Gamburd, and Sarnak to form an expander family. This conjecture has implications in both number theory (strengthening the affine linear sieve for Markoff numbers) and computational group theory (bounding runtime of the Product Replacement Algorithm for $SL_2(F_p)$). In this talk, we discuss recent progress toward proving connectivity of Markoff graphs and related results on Nielsen graphs of matrix pairs from $SL_2(F_p)$.
By Cruz Godar
By Jonathan Love
By Robert Corless
By Sébastien Labbé
By Ian Short
By Bernat Espigule
By Vlerë Mehmeti
By Karl-Mikael Perfekt
