We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3-bound syspi_1(X_{\PCH}) > 4/3 \log(g(X_{\PCH})). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.
We study how the systole of principal congruence coverings of a Hilbert modular variety grows when the degree of the covering goes to infinity. We prove that given a Hilbert modular variety $M$ of real dimension $2n$, the sequence of principal congruence coverings $M_{I}$ eventually satisfies $$sysπ_{1}(M_{I})\geq \frac{4}{3\sqrt{n}}\log(vol(M_{I}))-c,$$ where $c$ is a constant independent of $M_{I}$.
In this paper we prove that, for any arithmetic hyperbolic $n$-manifold $M$ of the first type, the systole of most of the principal congruence coverings $M_{I}$ satisfy $$sys_{1}(M_{I})\geq \frac{8}{n(n+1)}\log(vol(M_{I}))-c,$$ where $c$ is a constant independent of $I$. This generalizes previous work of Buser and Sarnak, and Katz, Schaps and Vishne in dimension 2 and 3. As applications, we obtain explicit estimates for systolic genus of hyperbolic manifolds studied by Belolipetsky and the distance of homological codes constructed by Guth and Lubotzky. In an appendix together with Cayo Dória w…
We examine the large systole problem, which concerns compact hyperbolic Riemannian surfaces whose systole, the length of the shortest noncontractible loops, grows logarithmically in genus. The generalization of a construction of Buser and Sarnak by Katz, Schaps, and Vishne, which uses principal "congruence" subgroups of a fixed cocompact arithmetic Fuchsian, achieves the current maximum known growth constant of γ= 4/3. We prove that this is the best possible value of γfor this construction using arithmetic Fuchsians in the congruence case. The final section compares the large systole problem w…
We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of "Bolza twins" (using magma). Like the Hurwitz triplets, these correspond to the factoring of certain rational primes in the ring of integers of the invariant trace field of the surface. We exploit random sampling combin…
In this paper we study the systole growth of arithmetic locally symmetric spaces up congruence covers and show that this growth is at least logarithmic in volume. This generalizes previous work of Buser and Sarnak as well as Katz, Schaps and Vishne where the case of compact hyperbolic 2- and 3-manifolds was considered.
Here we survey on the growth of systoles of arithmetic locally symmetric spaces under the congruence covering and give simple proofs for the best possible constants of Gromov for several important classes of symmetric spaces.
Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (globally) maximal surface $S_{max}$ a compact Riemann surface of genus $g$ whose systole has length $\mathop{msys}(g)$. In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prov…
The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are constructed by taking congruence covers of arithmetic surfaces. In this article we provide a new construction for a sequence of surfaces with systoles that grow logarithmically in their genera. We do this by combining a construction for graphs of large girth and a count of the number of $\mathrm{SL}_2(\mathbb{Z})$ matrices with positive entries and bounded trace…
In this article, we provide bounds on systoles associated to a holomorphic $1$-form $ω$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,ω)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $ω$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.
In our previous work we introduced, for a Riemannian surface $S$, the quantity $ Λ(S):=\inf_Fλ_0(F)$, where $λ_0(F)$ denotes the first Dirichlet eigenvalue of $F$ and the infimum is taken over all compact subsurfaces $F$ of $S$ with smooth boundary and abelian fundamental group. A result of Brooks implies $Λ(S)\geλ_0(\tilde{S})$, the bottom of the spectrum of the universal cover $\tilde{S}$. In this paper, we discuss the strictness of the inequality. Moreover, in the case of curvature bounds, we relate $Λ(S)$ with the systole, improving a result by the last named author.
The purpose this article is to try to understand the mysterious coincidence between the asymptotic behavior of the volumes of the Moduli Space of closed hyperbolic surfaces of genus $g$ with respect to the Weil-Petersson metric and the asymptotic behavior of the number of arithmetic closed hyperbolic surfaces of genus $g$. If the set of arithmetic surfaces is well distributed then its image for any interesting function should be well distributed too. We investigate the distribution of the function systole. We give several results indicating that the systoles of arithmetic surfaces can not be c…
The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds for the translation length of a semisimple element x in SL_n(R) in terms of its associated Mahler measure. We use these geometric methods to prove the existence of extensions of number fields in which fixed sets of primes have certain prescribed splitting behavior.
In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider some features of the forgetful map from the above moduli space of spherical surfaces with conical points to the associated moduli space of pointed Riemann surfaces, such as its properness, which follows from an explicit systole inequality that relates metric invariants (spherical systole) and conformal invariant (extremal systole).
We study the systole of a random surface, where by a random surface we mean a surface constructed by randomly gluing together an even number of triangles. We study two types of metrics on these surfaces, the first one coming from using ideal hyperbolic triangles and the second one using triangles that carry a given Riemannian metric. In the hyperbolic case we compute the limit of the expected value of the systole when the number of triangles goes to infinity (approximately 2.484). We also determine the asymptotic probability distribution of the number of curves of any finite length. This turns…
On the thick part of the moduli space of Riemann surfaces, where there is a positive lower bound of the systole of the surface, we show that all Weil-Petersson Riemannian curvatures are bounded, independent of the genus of the surface.
4-dimensional spaces equipped with 2-dimensional (complex holomorphic or real smooth) completely integrable distributions are considered. The integral manifolds of such distributions are totally null and totally geodesics 2-dimensional surfaces which are called the null strings. Properties of congruences (foliations) of such 2-surfaces are studied. Some relations between properties of congruences of null strings, Petrov-Penrose type of SD Weyl spinor and algebraic types of the traceless Ricci tensor are analyzed.
P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic invariant of a 4-manifold X with indefinite intersection form. The estimate turns out to be polynomial, rather than logarithmic, in χ(X), if the conjectured surjectivity of the period map is correct. Such surjectivity is targeted by the current work in gauge theory. The surjectivi…
In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comparable to $\frac{1}{g^2}$ in $g$. In the proof of the upper bound, for any constant $ε>0$, we will construct a closed Riemann surface of genus $g$ in any $ε$-thick part of moduli space such that it admits a pants decomposition whose boundary curves all have length equal to $ε$, a…
In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surface. We show that the spine systole is a proper function and has its global minima precisely on the extremal surfaces (those containing the biggest possible discs). We also study minimal spines, which are critical points for the length functional. We completely classify minimal s…
In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmüller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to $g$, and is comparable to $1$ with respect to $n$. Moreover, we also study the asymptotic behavior, as $g$ goes to infinity, of t…
We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing group is realized as a Fuchsian group and a convenient finite generating set is computed. The upper bound is derived from the traces of the generators. Some explicit computations, including ones for non-arithmetic surfaces, are given. We apply a result of Cosac and Dória to show that the systolic length grows logarithmically with respect to the genus.
