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Creators/Authors contains: "Martin, Daniel E"

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  1. (, Inventiones mathematicae)
    Abstract In 2021, Chen proved that the size of any connected component of the Markoff mod$$p$$ p graph is divisible by$$p$$ p . In combination with the work of Bourgain, Gamburd, and Sarnak, Chen’s result resolves a conjecture of Baragar for all but finitely many primes: the Markoff mod$$p$$ p graph is connected. In particular, strong approximation for Markoff triples holds for all but finitely many primes. We provide an alternative proof of Chen’s theorem. 
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    Free, publicly-accessible full text available August 1, 2026
  2. (, Advances in Geometry)
    Abstract For π“ž an imaginary quadratic ring, we compute a fundamental polyhedron in hyperbolic 3-space for the action of PE2(π“ž), the projective elementary subgroup of PSL2(π“ž). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for PE2(π“ž), show that it has infinite index and is its own normalizer in PSL2(π“ž), and split PSL2(π“ž) into a free product with amalgamation that has PE2(π“ž) as one of its factors. 
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  3. (, Journal of Number Theory)
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  4. (, Open Book Series)
    null (Ed.)
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