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We prove a 1999 conjecture of Veys, which says that the opposite of the log-canonical threshold is the only possible pole of maximal order of Denef and Loeser’s motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic . We apply similar methods to study the weight function on the Berkovich skeleton associated with a degeneration of Calabi–Yau varieties. Our results suggest that the weight function induces a flow on the nonarchimedean analytification of the degeneration towards the Kontsevich–Soibelman skeleton.
We show that the -adic eigencurve is smooth at classical weight points which are regular at and give a precise criterion for étaleness over the weight space at those points. Our approach uses deformations of Galois representations.
We prove that for any pair of nonnegative numbers with , the set of -dimensional -badly approximable vectors is winning for Schmidt’s game. As a consequence, we give a direct proof of Schmidt’s conjecture using his game.
Let be a probability measure on with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of . We show that almost every sample path of the random walk on , when realized in Culler and Vogtmann’s outer space, converges to the simplex of a free, arational tree. We then prove that the space of simplices of free and arational trees, equipped with the hitting measure, is the Poisson boundary of . Using Bestvina and Reynolds’s and Hamenstädt’s description of the Gromov boundary of the complex of free factors of , this gives a new proof of the fact, due to Calegari and Maher, that the realization in of almost every sample path of the random walk converges to a boundary point. We get in addition that , equipped with the hitting measure, is the Poisson boundary of .
We explore the dynamics of the action of the mapping class group in genus on the -character variety. We prove that this action is ergodic on the connected components of Euler class , as it was conjectured by Goldman. In the connected component of Euler class there are two invariant open subsets; on one of them the action is ergodic. In this process we give a partial answer to a question posed by Bowditch.