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Item Infinitely many arithmetic alternating links(Mathematical Sciences Publishers, 2023) Baker, Mark D.; Reid, Alan W.Item Johnson–Schwartzman gap labelling for ergodic Jacobi matrices(EMS Press, 2023) Damanik, David; Fillman, Jake; Zhang, ZhengheWe consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphism on a compact metric space. Given an ergodic probability measure, we study the topological structure of the associated almost sure spectrum. We establish a gap labelling theorem in the spirit of Johnson and Schwartzman. That is, we show that the constant value the integrated density of states takes in a gap of the spectrum must belong to the countable Schwartzman group of the base dynamics. This result is a natural companion to a recent result of Alkorn and Zhang, which established a Johnson-type theorem for the families of Jacobi matrices in question.Item Totally geodesic hyperbolic 3-manifolds in hyperbolic link complements of tori in S4(Mathematical Sciences Publishers, 2023) Chu, Michelle; Reid, Alan W.Item G-valued crystalline deformation rings in the Fontaine–Laffaille range(Cambridge University Press, 2023) Booher, Jeremy; Levin, BrandonLet G be a split reductive group over the ring of integers in a ppp-adic field with residue field FF\mathbf {F}. Fix a representation ¯ρρ¯¯¯\overline {\rho } of the absolute Galois group of an unramified extension of QpQp\mathbf {Q}_p, valued in G(F)G(F)G(\mathbf {F}). We study the crystalline deformation ring for ¯ρρ¯¯¯\overline {\rho } with a fixed ppp-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for GGG-valued representations. In particular, we give a root theoretic condition on the ppp-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.Item Pseudo-Anosov subgroups of general fibered 3–manifold groups(American Mathematical Society, 2023) Leininger, Christopher; Russell, JacobWe show that finitely generated and purely pseudo-Anosov subgroups of fundamental groups of fibered 3–manifolds with reducible monodromy are convex cocompact as subgroups of the mapping class group via the Birman exact sequence. Combined with results of Dowdall–Kent–Leininger and Kent–Leininger–Schleimer, this establishes the result for the image of all such fibered 3–manifold groups in the mapping class group.Item The p-widths of a surface(Springer Nature, 2023) Chodosh, Otis; Mantoulidis, ChristosThe p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the p-widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets. We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the p-widths of the round sphere are attained by ⌊p–√⌋ great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be π−−√.En route to calculating the p-widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero.Item Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds(Springer Nature, 2023) Bader, Uri; Fisher, David; Miller, Nicholas; Stover, MatthewFor n≥2, we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps between complex hyperbolic manifolds, which is related to a question of Siu, that certain hyperbolic 3-manifolds cannot be totally geodesic submanifolds of complex hyperbolic manifolds, and that arithmeticity of complex hyperbolic manifolds is detected purely by the topology of the underlying complex variety, which is related to a question of Margulis. Our results also provide some evidence for a conjecture of Klingler that is a broad generalization of the Zilber–Pink conjecture.Item End-periodic homeomorphisms and volumes of mapping tori(Wiley, 2023) Field, Elizabeth; Kim, Heejoung; Leininger, Christopher; Loving, MarissaGiven an irreducible, end-periodic homeomorphism f:S→S$f: S \rightarrow S$ of a surface with finitely many ends, all accumulated by genus, the mapping torus, Mf$M_f$, is the interior of a compact, irreducible, atoroidal 3-manifold M¯f$øverlineM_f$ with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of M¯f$øverlineM_f$ in terms of the translation length of f$f$ on the pants graph of S$S$. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotically sharp.Item Some remarks on group actions on hyperbolic 3-manifolds(University of Primorska, 2022) Reid, Alan; Salgueiro, AntonioWe prove that there are infinitely many non-commensurable closed orientable hyperbolic 3-manifolds X, with the property that there are finite groups G1 and G2 acting freely by orientation-preserving isometries on X with X/G1 and X/G2 isometric, but G1 and G2 are not conjugate in Isom(X). We provide examples where G1 and G2 are non-isomorphic, and prove analogous results when G1 and G2 act with fixed-points.Item A universal Cannon-Thurston map and the surviving curve complex(American Mathematical Society, 2022) Gültepe, Funda; Leininger, Christopher; Pho-on, WitsarutUsing the Birman exact sequence for pure mapping class groups, we construct a universal Cannon-Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. Along the way we prove hyperbolicity of this complex and identify its boundary as a space of laminations. As a corollary we obtain a universal Cannon-Thurston map to the boundary of the ordinary curve complex, extending earlier work of the second author with Mj and Schleimer [Comment. Math. Helv. 86 (2011), pp. 769–816].Item Orthogonal rational functions with real poles, root asymptotics, and GMP matrices(American Mathematical Society, 2023) Eichinger, Benjamin; Lukić, Milivoje; Young, GiorgioThere is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for . We extend aspects of this theory in the setting of rational functions with poles on, obtaining a formulation which allows multiple poles and proving an invariance with respect to preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.Item Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent(Springer Nature, 2023) Avila, Artur; Damanik, David; Zhang, ZhengheWe consider discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant Hölder continuous function defined on a subshift of finite type with a fully supported ergodic measure admitting a local product structure and a fixed point, then the Lyapunov exponent is positive away from a discrete set of energies. Moreover, for sampling functions in a residual subset of the space of Hölder continuous functions, the Lyapunov exponent is positive everywhere. If we consider locally constant or globally fiber bunched sampling functions, then the Lyapuonv exponent is positive away from a finite set. Moreover, for sampling functions in an open and dense subset of the space in question, the Lyapunov exponent is uniformly positive. Our results can be applied to any subshift of finite type with ergodic measures that are equilibrium states of Hölder continuous potentials. In particular, we apply our results to Schrödinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains.Item Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence(De Gruyter, 2022) Frei, Sarah; Hassett, Brendan; Várilly-Alvarado, AnthonyGiven a smooth projective variety over a number field and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class. We are interested in the case of K3 surfaces. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial after specialization at a set of places of positive natural density. We deduce that there exist cubic fourfolds over number fields that are conjecturally irrational, with rational reduction at a positive proportion of places. We also deduce that there are twisted derived equivalent K3 surfaces which become derived equivalent after reduction at a positive proportion of places.Item Hierarchical hyperbolicity of graph products(EMS Press, 2022) Berlyne, Daniel; Russell, JacobWe show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups.We also answer two questions of Genevois about the geometry of the electrification of a graph product of finite groups.Item Zero measure spectrum for multi-frequency Schrödinger operators(EMS Press, 2022) Chaika, Jon; Damanik, David; Fillman, Jake; Gohlke, PhilippBuilding on works of Berthé–Steiner–Thuswaldner and Fogg–Nous, we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence, we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schrödinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.Item Explicit computation of symmetric differentials and its application to quasihyperbolicity(Mathematical Science Publishers, 2022) Bruin, Nils; Thomas, Jordan; Várilly-Alvarado, AnthonyWe develop explicit techniques to investigate algebraic quasihyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth’s sextic surface, apart from some well-known ones, must pass through at least four singularities, and that genus 1 curves must pass through at least two. On the surface classifying perfect cuboids, our methods show that rational curves, again apart from some well-known ones, must pass through at least seven singularities, and that genus 1 curves must pass through at least two. We also improve lower bounds on the dimension of the space of symmetric differentials on surfaces with A1-singularities, and use our work to show that Barth’s decic, Sarti’s surface, and the surface parametrizing 3×3 magic squares of squares are all algebraically quasihyperbolic.Item Absolutely continuous spectrum for CMV matrices with small quasi-periodic Verblunsky coefficients(American Mathematical Society, 2022) Li, Long; Damanik, David; Zhou, QiWe consider standard and extended CMV matrices with small quasi-periodic Verblunsky coefficients and show that on their essential spectrum, all spectral measures are purely absolutely continuous. This answers a question of Barry Simon from 2005.Item Hyperbolic cone metrics and billiards(Elsevier, 2022) Erlandsson, Viveka; Leininger, Christopher J.; Sadanand, ChandrikaA negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.Item Must the Spectrum of a Random Schrödinger Operator Contain an Interval?(Springer Nature, 2022) Damanik, David; Gorodetski, AntonWe consider Schrödinger operators in ℓ2(Z) whose potentials are given by independent (not necessarily identically distributed) random variables. We ask whether it is true that almost surely its spectrum contains an interval. We provide an affirmative answer in the case of random potentials given by a sum of a perturbatively small quasi-periodic potential with analytic sampling function and Diophantine frequency vector and a term of Anderson type, given by independent identically distributed random variables (with some small-gap assumption for the support of the single-site distribution). The proof proceeds by extending a result about the presence of ground states for atypical realizations of the classical Anderson model, which we prove here as well and which appears to be new.Item Absence of absolutely continuous spectrum for generic quasi-periodic Schrödinger operators on the real line(Springer Nature, 2022) Damanik, David; Lenz, DanielWe show that a generic quasi-periodic Schrödinger operator in L2(ℝ) has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely continuous spectrum.