Abstract Let W⊂ P 1× P 1× P 1 be a surface given by the vanishing of a (2, 2, 2)-form. These
surfaces admit three involutions coming from the three projections W→ P 1× P 1, so we call …

We confirm, for the primes up to 3000, the conjecture of Bourgain-Gamburd-Sarnak and
Baragar on strong approximation for the Markoff surface x 2+ y 2+ z 2= 3 xyz modulo primes …

We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible
number of nodes outside the 'giant component'and on the size of individual connected …

For each prime p, we study the eigenvalues of a 3-regular graph on roughly p 2 vertices
constructed from the Markoff surface. We show they asymptotically follow the Kesten–McKay …

The Markoff group of transformations is a group Γ of affine integral morphisms, which is
known to act transitively on the set of all positive integer solutions to the equation x 2+ y 2+ z …

Charles, Goren, and Lauter [J. Cryptology 22 (1), 2009] explained how one can construct
hash functions using expander graphs in which it is hard to find paths between specified …

We prove the non-planarity of a family of 3-regular graphs constructed from the solutions to
the Markoff equation x2 C y2 C z2 D xyz modulo prime numbers greater than 7. The proof …

If G is a reductive algebraic group over Z, the G-character variety of a finitely presented
group F parameterizes the set of closed conjugation orbits in Hom (F, G). The group of …

We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible
number of nodes outside the" giant component" and on the size of individual connected …

We obtain a new bound for the number of solutions to polynomial equations in cosets of
multiplicative subgroups in finite fields, which generalizes previous results of P. Corvaja and …