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We give a complete condition for any $n$ elements of ${\rm PSL}(2, {\mathbb R})$ to generate a Fuchsian group which is a Schottky group on that set of generators, and apply it to a question of Bowditch on representations of surface groups into ${\rm PSL}(2, {\mathbb R})$.

2000 Mathematical Subject Classification: 20H10, 32G15.

Given the Diophantine equation a2+b2+c2=3abc, a solution triple of natural numbers (a, b, c) can be arranged in ascending order so that a[les ]b[les ]c. Then, given the largest element c, one can ask whether this uniquely determines the triple. This is referred to as the Markoff conjecture. The paper proves that, if c is prime, then there is indeed only one triple that solves the equation with c as the largest element. The proof uses only standard algebraic number theory, but it was prompted by geometric considerations.

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MARKED SCHOTTKY GROUPS AND REPRESENTATIONS OF SURFACE GROUPS

THE UNIQUENESS OF THE PRIME MARKOFF NUMBERS

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MARKED SCHOTTKY GROUPS AND REPRESENTATIONS OF SURFACE GROUPS

THE UNIQUENESS OF THE PRIME MARKOFF NUMBERS