Cubic Surfaces with a Discrete Automorphism Group

We will present results from a recent preprint by János Kollár and David Villalobos-Paz (https://arxiv.org/pdf/2410.03934) concerning the automorphisms of affine cubic surfaces. Specifically, the involutions corresponding to singular points of the boundary curve form a free product and generate the automorphism group up to a finite index. The most famous example is the Markov surface given by $x^2+y^2+z^2=xyz$. We will also examine the case of equations of the form $xyz = g(x,y)$ and present some related results.