An almost linear time algorithm testing whether the Markoff graph modulo $p$ is connected

Abstract

The Markoff graph modulo $p$ is known to be connected for all but finitely many primes $p$ (see Eddy, Fuchs, Litman, Martin, Tripeny, and Vanyo [arxiv:2308.07579]), and it is conjectured that these graphs are connected for all primes. In this paper, we provide an algorithmic realization of the process introduced by Bourgain, Gamburd, and Sarnak [arxiv:1607.01530] to test whether the Markoff graph modulo $p$ is connected for arbitrary primes. Our algorithm runs in $o(p^{1 + ε})$ time for every $ε> 0$. We demonstrate this algorithm by confirming that the Markoff graph modulo $p$ is connected for all primes less than one million.

Figures (7)

• Figure 1: The Markoff tree $\mathcal{G^\times}$.
• Figure 2: The Markoff graph $\mathcal{G}^\times_5$.
• Figure 3: The factor trie for $n = 60$.
• Figure 4: Total number of bad triples for all primes less than $1,000,000$ (top) and random primes less than $110,000,000$ (bottom).
• Figure 5: Maximum number of orbits checked during any iteration of Step \ref{['step:cosets']} for all primes less than $1,000,000$ (top) and random primes less than $110,000,000$ (bottom).

Theorems & Definitions (29)

• Lemma 1
• proof
• Conjecture 1
• Conjecture 2
• Lemma 2
• proof
• Theorem 1
• Lemma 3
• proof
• Lemma 4