Congruence images of continued-fraction semigroups and trace residues

Abstract

Let [Formula: see text] be a finite alphabet with at least two elements, and let [Formula: see text] be the continued-fraction semigroup generated by the matrices [Formula: see text]. Its even-length words form a determinant-one subsemigroup [Formula: see text]. Writing [Formula: see text] for the greatest common divisor of the differences of the elements of [Formula: see text], we prove that the reduction of [Formula: see text] modulo [Formula: see text] equals [Formula: see text] exactly when [Formula: see text] is coprime to [Formula: see text]. Thus the only congruence-image obstructions occur at primes dividing [Formula: see text]. For good moduli, the trace residues of even words cover every residue class. For bad moduli, the image is cyclic and the trace residues are controlled by a Lucas recurrence. In particular, for the alphabet [Formula: see text], which is central in the sharp bounded-partial-quotient form of McMullen’s conjecture, no finite congruence obstruction exists for the corresponding Pell trace problem. We also record consequences for fixed-modulus counting, finite-state distribution, Pell trace congruences, and square-free divisibility away from the exceptional primes. This paper is purely local and does not claim a trace local–global theorem.

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