More Like this

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. 

   more » « less
2. Blown-up toric surfaces with non-polyhedral effective cone

   https://doi.org/10.1515/crelle-2023-0022  

   Castravet, Ana-Maria; Laface, Antonio; Tevelev, Jenia; Ugaglia, Luca (May 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We construct projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone.As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space ${\overline{M}}_{0,n}$\overline{M}_{0,n}of stable rational curves is not polyhedral for $n\ge 10$n\geq 10.These results hold both in characteristic 0 and in characteristic 𝑝, for all primes 𝑝.Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order.Our analysis relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang–Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic 0 and in characteristic 𝑝, for an infinite set of primes 𝑝 of positive density. 

   more » « less
3. Periods of elliptic surfaces with $$p_g=q=1$$

   https://doi.org/10.1017/fms.2024.85  

   Engel, Philip; Greer, François; Ward, Abigail (November 2024, Forum of Mathematics, Sigma)

   Abstract We prove that the period mapping is dominant for elliptic surfaces over an elliptic curve with$$12$$nodal fibers, and that its degree is larger than$$1$$. This settles the final case of infinitesimal Torelli for a generic elliptic surface. 

   more » « less
4. Optimal (Euclidean) Metric Compression

   https://doi.org/10.1137/20M1371324  

   Indyk, P.; Wagner, T. (January 2022, SIAM journal on computing)

   We study the problem of representing all distances between n points in Rd, with arbitrarily small distortion, using as few bits as possible. We give asymptotically tight bounds for this problem, for Euclidean metrics, for ℓ1 (a.k.a.~Manhattan) metrics, and for general metrics. Our bounds for Euclidean metrics mark the first improvement over compression schemes based on discretizing the classical dimensionality reduction theorem of Johnson and Lindenstrauss (Contemp.~Math.~1984). Since it is known that no better dimension reduction is possible, our results establish that Euclidean metric compression is possible beyond dimension reduction. 

   more » « less
5. Patterson–Sullivan measures for transverse subgroups

   https://doi.org/10.3934/jmd.2024009  

   Canary, Richard; Zhang, Tengren; Zimmer, Andrew (January 2024, Journal of Modern Dynamics)

   Forni, Giovanni (Ed.)

   We study Patterson–Sullivan measures for a class of discrete subgroups of higher rank semisimple Lie groups, called transverse groups, whose limit set is well-defined and transverse in a partial flag variety. This class of groups includes both Anosov and relatively Anosov groups, as well as all discrete subgroups of rank one Lie groups. We prove an analogue of the Hopf–Tsuji–Sullivan dichotomy and then use this dichotomy to prove a variant of Burger's Manhattan Curve Theorem. We also use the Patterson–Sullivan measures to obtain conditions for when a subgroup has critical exponent strictly less than the original transverse group. These gap results are new even for Anosov groups. 

   more » « less