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1. Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

   https://doi.org/10.1515/crelle-2022-0035  

   Bray, Harrison; Canary, Richard; Kao, Lien-Yung; Martone, Giuseppe (August 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We show that if an eventually positive, non-arithmetic, locally Hölder continuous potential for a topologically mixingcountable Markov shift with (BIP) has an entropy gap at infinity,then one may apply the renewal theorem of Kesseböhmer and Kombrink to obtain counting and equidistributionresults. We apply these general results to obtain counting and equidistribution results for cusped Hitchinrepresentations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups. 

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2. On the profinite rigidity of triangle groups

   M. R. Bridson, D. B. (January 2020, Preprint)

   We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method based on character varieties that can be used to distinguish between the profinite completions of certain groups. 

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3. Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

   Casale, Guy; Freitag, James; Nagloo, Joel (November 2020, Annals of mathematics)

   null (Ed.)

   We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem that have interesting consequences. In particular, we apply our results to give a complete proof of an assertion of Painlevé (1895). We also answer certain cases of the André-Pink conjecture, namely, in the case of orbits of commensurators of Fuchsian groups. 

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4. The fundamental inequality for cocompact Fuchsian groups

   https://doi.org/10.1017/fms.2022.94  

   Kosenko, Petr; Tiozzo, Giulio (January 2022, Forum of Mathematics, Sigma)

   Abstract We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups. 

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5. Testing a Unified Model of Arithmetic

   Braithwaite, David W.; Siegler, Robert S. (January 2022, Proceedings of the Annual Conference of the Cognitive Science Society)

   We describe UMA (Unified Model of Arithmetic), a theory of children’s arithmetic implemented as a computational model. UMA extends a theory of fraction arithmetic (Braithwaite et al., 2017) to include arithmetic with whole numbers and decimals. We evaluated UMA in the domain of decimal arithmetic by training the model on problems from a math textbook series, then testing it on decimal arithmetic problems that were solved by 6th and 8th graders in a previous study. UMA’s test performance closely matched that of children, supporting three assumptions of the theory: (1) most errors reflect small deviations from standard procedures, (2) between-problem variations in error rates reflect the distribution of input that learners receive, and (3) individual differences in strategy use reflect underlying variation in learning parameters. 

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