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Award ID contains: 2336000

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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 𝓞 an imaginary quadratic ring, we compute a fundamental polyhedron in hyperbolic 3-space for the action of PE2(𝓞), the projective elementary subgroup of PSL2(𝓞). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for PE2(𝓞), show that it has infinite index and is its own normalizer in PSL2(𝓞), and split PSL2(𝓞) into a free product with amalgamation that has PE2(𝓞) as one of its factors. 
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    Free, publicly-accessible full text available January 1, 2026
  3. ; ; ; ; (, Proceedings of the London Mathematical Society)
    Abstract Markoff mod‐ graphs are conjectured to be connected for all primes . In this paper, we use results of Chen and Bourgain, Gamburd, and Sarnak to confirm the conjecture for all . We also provide a method that quickly verifies connectivity for many primes below this bound. In our study of Markoff mod‐ graphs, we introduce the notion ofmaximal divisorsof a number. We prove sharp asymptotic and explicit upper bounds on the number of maximal divisors, which ultimately improves the Markoff graph ‐bound by roughly 140 orders of magnitude as compared with an approach using all divisors. 
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