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Register a new userLow-lying Geodesics in an Arithmetic Hyperbolic Three-Manifold
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Katie McKeon
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We examine closed geodesics in the quotient of hyperbolic three space by the discrete group of isometries SL(2,Z[i]). There is a correspondence between closed geodesics in the manifold, the complex continued fractions originally studied by Hurwitz, and binary quadratic forms over the Gaussian integers. According to this correspondence, a geodesic is called fundamental if the associated binary quadratic form is. Using techniques from sieve theory, symbolic dynamics, and the theory of expander graphs, we show the existence of a compact set in the manifold containing infinitely many fundamental geodesics.
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On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the volume. By a novel combination of diophantine and geometric arguments in a noncommutative setting, we obtain bounds as strong as the best corresponding results on arithmetic surfaces.
Froyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in three and four-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal $L$-spaces. In particular, we study relations between Froyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive effective upper bounds for the Froyshov invariants of minimal hyperbolic $L$-spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Froyshov invariants of minimal $L$-spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Froyshov invariants for all spin$^c$ structures on the Seifert-Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 le r le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
We prove that any arithmetic hyperbolic $n$-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic $(n+1)$-manifold or its universal $mathrm{mod}~2$ Abelian cover can.
Let $t in mathbb{N}$, $eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q leq x^{5/12-eta}$, $q$ not a multiple of the conductor of the exceptional character $chi^*$ (if it exists). Suppose further that, [ max {p : p | q } < exp (frac{log x}{C log log x}) ; ; {and} ; ; prod_{p | q} p < x^{delta}, ] where $C$ and $delta$ are suitable positive constants depending on $t$ and $eta$. Let $a in mathbb{Z}$, $(a,q)=1$ and [ mathcal{A} = {n in (x/2, x]: n equiv a pmod{q} } . ] We prove that there are primes $p_1 < p_2 < ... < p_t$ in $mathcal{A}$ with [ p_t - p_1 ll qt exp (frac{40 t}{9-20 theta}) . ] Here $theta = (log q) / log x$.
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