Sharp vertex connectivity of the Markoff graphs modulo $p$

Abstract

For a prime $p$ and $k\in {\mathbb{F}}_p$, the generalized Markoff graph $G_{p,k}$ is an undirected graph whose vertices are the solutions over the finite field ${\mathbb{F}}_p$ of the normalized Markoff equation \[ x_1^2+x_2^2+x_3^2=x_1x_2x_3+k, \] where two vertices are adjacent if they differ by a Vieta involution. The Markoff graph $G_p$ is obtained from $G_{p,0}$ by removing the origin. The structure of $G_p$ has been the subject of extensive study; in particular, a major breakthrough of Bourgain, Gamburd, and Sarnak established that $G_p$ contains a giant connected component. Combined with Chen's remarkable divisibility theorem, this implies that $G_p$ is connected for all sufficiently large primes $p$. In the same paper, Bourgain, Gamburd, and Sarnak further asked whether the family $\{G_p:\text{primes }p\ge 5\}$ forms an expander family. This motivates us to investigate the robustness of connectivity in the Markoff graphs. The main result of this paper is proved in the general setting: for every prime $p\ge 5$ and every $k\in {\mathbb{F}}_p\setminus \{4\}$, each connected component $C$ of $G_{p,k}$ with $|V(C)|\ge 3$ is $2$-connected. Reducing to the case $k=0$, we conclude that if the Markoff graph $G_p$ is connected, then it is in fact $2$-connected. Consequently, the Markoff graph $G_p$ is $2$-connected for all sufficiently large primes $p$. This is sharp in the sense that $G_p$ is not $3$-connected for any prime $p\ge 7$.