Hyperbolic manifolds with a large number of systoles

The hyperbolic $n$-space is the complete simply connected $n$-dimensional Riemannian manifold with constant sectional curvature equal to $-1$. The hyperboloid model of the hyperbolic $n$-space is given by

with the metric $ds^{2}=-dx_{0}^{2}+dx_{1}^{2}+\cdots+dx_{n}^{2}$.

The identity component $\mathrm{SO}(n,1)^{\circ}$ of the Lie group $\mathrm{SO}(n,1)$ is isomorphic to the orientation preserving isometries of $\mathbb{H}^{n}$. Given a lattice $\Gamma\subset\mathrm{Isom}^{+}(\mathbb{H}^{n})$, i.e, a discrete subgroup having finite covolume with respect to the Haar measure, the associated quotient space $M=\Gamma\backslash\mathbb{H}^{n}$ is a finite volume hyperbolic orbifold. It is a manifold whenever $\Gamma$ is torsion-free.

We recall that an element $\gamma$ in $\mathrm{Isom}^{+}(\mathbb{H}^{n})$ is called

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  elliptic, if it has a fixed point on $\mathbb{H}^{n}$.

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  parabolic, if it has exactly one fixed point in $\partial\mathbb{H}^{n}$.

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  loxodromic, if it has two fixed points in $\partial\mathbb{H}^{n}$.

The displacement at $x\in\mathbb{H}^{n}$ of a loxodromic element $\gamma$ is defined by $l(\gamma,x):=d(x,\gamma x)$. The displacement of $\gamma$ (also called translation length) is defined by

A systole of a hyperbolic orbifold $M=\Gamma\backslash\mathbb{H}^{n}$ is any closed geodesic of shortest length in $M$, and its length is denoted by $\mathrm{sys}(M)$. The kissing number $\mathrm{Kiss}(M)$ is defined as the number of free homotopy classes of oriented closed geodesics in $M$ that realize $\mathrm{sys}(M)$. It is well known that when $M$ is a manifold, there exists a one-to-one correspondence between parametrized closed geodesic in $M$ up to unit speed reparametrization and conjugacy classes of loxodromic elements in $\Gamma$. Moreover, the length of the closed geodesic corresponds to the translation length of the loxodromic element. This relation allows to study $\mathrm{sys}(M)$ and $\mathrm{Kiss}(M)$ through the number of conjugacy classes of loxodromic elements in $\Gamma$ that realize $\mathrm{sys}(M)$ as their translation length.

In this section we will recall the construction of spinor groups. These are the algebraic groups associated to fundamental groups of arithmetic hyperbolic orbifolds. For further details, we refer the reader to [Mur19, Section 2], and the references therein.

Let $k$ be a field with char $k\neq 2$, $E$ a $n$-dimensional vector space over $k$, and $f$ a non-degenerate quadratic form on $E$ with associated bilinear form $\Phi$. The Clifford algebra of $f$, denoted by $\mathscr{C}(f,k)$, is a unitary associative algebra over $k$ given by the quotient $\mathscr{C}(f,k)=T(E)/\mathfrak{a}_{f}$, where $T(E)$ denotes the tensor algebra of $E$, and $\mathfrak{a}_{f}$ is the two-sided ideal of $T(E)$ generated by the elements $x\otimes y+y\otimes x-2\Phi(x,y)$. If we choose an orthogonal basis $e_{1},\dots,e_{n}$ of $E$ with respect to $f,$ we have in $\mathscr{C}(f,k)$ the relations $e_{v}^{2}=f(e_{v})$ and $e_{v}e_{\mu}=-e_{\mu}e_{v}$ for $\mu,v\in\{1,\ldots,n\},\ \mu\neq v.$ Let $\mathscr{P}_{n}$ be the power set of $\{1,...,n\}$. For $M=\{\mu_{1},\dots,\mu_{\nu}\}\in\mathscr{P}_{n}$ with $\mu_{1}<\dots<\mu_{\nu},$ we define $e_{M}=e_{\mu_{1}}\cdot...\cdot e_{\mu_{\nu}}$, where we adopt the convention $e_{\emptyset}=1$. The $2^{n}$ elements $e_{M}$, $M\in\mathscr{P}_{n}$ determine a basis for $\mathscr{C}(f,k)$. Hence, any $s\in\mathscr{C}(f,k)$ is written uniquely as

with $s_{\mathbb{R}},s_{M}\in k$. We call the coefficient $s_{\mathbb{R}}$ as the real part of $s$. We identify $k$ with $k\cdot 1$, and $E$ with the $k$-linear subspace in $\mathscr{C}(f,k)$ generated by $e_{1},e_{2},\ldots,e_{n}$. The algebra $\mathscr{C}(f,k)$ has an anti-involution $*$. On the elements $e_{M}$ this map acts by $e_{M}^{*}=(-1)^{\nu(\nu-1)/2}\ e_{M}$, where $\nu=|M|$. The span of the elements $e_{M}$ with $|M|$ even is a subalgebra $\mathscr{C}^{+}(f)$ of $\mathscr{C}(f),$ called the even Clifford subalgebra of $f$. The spin group of $f$ is defined as

In the case $k=\mathbb{R}$, $E=\mathbb{R}^{n+1}$ and $f=-x_{0}^{2}+x_{1}^{2}+\cdots+x_{n}^{2}$, the corresponding spin group is denoted by $\mathrm{Spin}(n,1)$. For an element $s\in\mathrm{Spin}(n,1)$ the linear map $\varphi_{s}:E\to E$ given by $\varphi_{s}(x)=sxs^{*}$ lies in $\mathrm{SO}(n,1)^{\circ}$, and the map $s\to\varphi_{s}$ is a two-sheeted covering of $\mathrm{SO}(n,1)^{\circ}$, with kernel $\{\pm 1\}$. Since the image of a lattice under a finite covering map is also a lattice, in order to produce hyperbolic orbifolds we will implicitly contruct lattices in $\mathrm{Spin}(n,1)$ and project them to $\mathrm{SO}(n,1)^{\circ}$. Furthermore, we will abuse terminology saying that an element $s\in\mathrm{Spin}(n,1)$ is elliptic (resp. parabolic or loxodromic) if $\varphi_{s}$ is elliptic (resp. parabolic or loxodromic).

We recall that a discrete subgroup $\Gamma\subset\mathrm{Spin}(1,n)$ is arithmetic if there exist a number field $k$, a $k$-algebraic group H, and an epimorphism $\varphi:\mathrm{\textbf{H}}(k\otimes_{\mathbb{Q}}\mathbb{R})\rightarrow\mathrm{Spin}(1,n)$ with compact kernel such that $\varphi(\mathrm{\textbf{H}}(\mathcal{O}_{k}))$ is commensurable to $\Gamma$. Here, $\mathcal{O}_{k}$ denotes the ring of integers of $k$ and $\mathrm{\textbf{H}}(\mathcal{O}_{k})=\mathrm{\textbf{H}}\cap\mathrm{GL}_{n}(\mathcal{O}_{k})$ with respect to some fixed embedding of H into $\mathrm{GL}_{n}$. The field $k$ is the field of definition of $\Gamma$. Any algebraic $k$-group H satisfying these properties is called admissible. A hyperbolic orbifold $M=\Gamma\backslash\mathbb{H}^{n}$ such that $\Gamma$ is an arithmetic subgroup of $\mathrm{Isom}^{+}(\mathbb{H}^{n})$ is called an arithmetic hyperbolic orbifold. It follows from Borel and Harish–Chandra’s Theorem that any arithmetic hyperbolic orbifold has finite volume [BHC62].

The admissibility condition implies that $k$ is a totally real number field, H is a simply-connected algebraic $k$-group, and by fixing an embedding $k\subset\mathbb{R}$ we can assume that $\mathrm{\textbf{H}}(\mathbb{R})=\mathrm{Spin}(n,1)$ (see [BE12, Sec. 2.2] and [Eme09, Sec. 13.1]). Suppose $k$ is a totally real number field, and $f$ is a quadratic form over $k$. The admissibility of the algebraic $k$-group $\mathrm{\textbf{H}}=\mathrm{Spin}_{f}$ is equivalent to the fact that $f$ has signature $(n,1)$ over $\mathbb{R}$, and $f^{\sigma}$ is definite for any non-trivial embedding $\sigma:k\rightarrow\mathbb{R}$, where $f^{\sigma}$ is the quadratic form defined on the same $k$-space of $f$, given by $f^{\sigma}(x)=\sigma(f(x))$. A quadratic form satisfying these conditions will be called an admissible quadratic form. The arithmetic groups commensurable to $\mathrm{Spin}_{f}(\mathcal{O}_{k})$ with $f$ admissible, are called arithmetic groups of the first type. For $n$ even, any arithmetic subgroup of $\mathrm{Spin}(n,1)$ is of the first type. There is a second class in odd dimensions, arising from skew-hermitian forms over division quaternion algebras. For $n=7$ there is a third class, arising from certain Cayley algebras. In this work we will deal with arithmetic hyperbolic orbifolds of the first type in dimensions $n\geq 2$.

Let $\Gamma$ be an arithmetic subgroup of $\mathrm{Spin}(n,1)$ commensurable with $\varphi(\mathrm{\textbf{H}}(\mathcal{O}_{k}))$, and $M=\Gamma\backslash\mathbb{H}^{n}$ the corresponding hyperbolic arithmetic orbifold. If $I\subset\mathcal{O}_{k}$ is a non-zero ideal of $\mathcal{O}_{k}$, the principal congruence subgroup of $\Gamma$ associated to $I$ is the subgroup $\Gamma(I):=\Gamma\cap\varphi(\mathrm{\textbf{H}}(I))$, where

and $\pi_{I}$ denotes the reduction modulo $I$ map. The associated principal congruence covering is $M_{I}=\Gamma(I)\backslash\mathbb{H}^{n}\rightarrow M$. Since $\Gamma(I)$ is a normal finite-index subgroup of $\Gamma$, the covering $M_{I}\rightarrow M$ is a regular finite-sheeted covering map. More generally, a discrete subgroup $\Lambda<\Gamma$ is called a congruence subgroup if $\Gamma(I)\subset\Lambda$ for some ideal $I\subset\mathcal{O}_{k}$.

Let $f$ be an admissible quadratic form over a totally real number field $k$ of degree $d$. We can describe the group $\Gamma=\mathrm{Spin}_{f}(\mathcal{O}_{k})$ and its principal congruence subgroups $\Gamma(I)$ in the following way. Denote by $e_{1},e_{2},\ldots,e_{n+1}$ an orthogonal basis with respect to $f$. Then under the linear representation given by left multiplication in $\mathscr{C}^{+}(f,\mathbb{R})$ we get

and

(see [Mur19, Sec. 2.4]). To simplify the discussion, we will denote by $\mathcal{Q}$ the $\mathcal{O}_{k}$-order in $\mathscr{C}^{+}(f,\mathbb{R})$ given by

For a principal ideal $I=(\alpha)$, $\alpha\in\mathcal{O}_{k}$, we denote $\Gamma(I)$ simply by $\Gamma(\alpha)$. In the same way, we denote by $\mathrm{N}(\alpha)$ the norm of the ideal $(\alpha)$. We will present a series of results relating the real part of elements in $\Gamma(\alpha)$ with $\alpha$ that will be necessary in the course of the investigation.

The following complement of Lemma 4.1 will also be useful.

An important example of congruence subgroup which will play a special role in this work is the following. Fix an element $\alpha\in\mathcal{O}_{k}$. Suppose that $\tau\in(\mathcal{O}_{k}/\alpha\mathcal{O}_{k})^{\times}$ is an element of order 2, and define

The group $\Gamma_{\tau}(\alpha)$ is a normal subgroup of $\Gamma$. Indeed, let $t\in\mathcal{O}_{k}$ such that $\tau=t+\alpha\mathcal{O}_{k}$. We note that $\{1+\alpha\mathcal{Q},t+\alpha\mathcal{Q}\}$ is a central subgroup of $\left(\mathcal{Q}/\alpha\mathcal{Q}\right)^{\times}$, hence $\Gamma_{\tau}(\alpha)$ is normal since it is the preimage of this normal subgroup under the natural projection map $\Gamma\to\left(\mathcal{Q}/\alpha\mathcal{Q}\right)^{\times}$ .

Let $k$ be a totally real number field, and $(E,f)$ be an admissible $(n+1)$-dimensional quadratic space over $k$ (see subsection 4.2). Since $f$ has signature $(n,1)$ and $k\subset\mathbb{R}$, by the Gram-Schmidt process and the Law of Inertia, there exists a basis $\{e_{0},e_{1},\ldots,e_{n}\}$ of $E$ such that in this basis $f$ has the diagonal form $f=-a_{0}x_{0}^{2}+a_{1}x_{1}^{2}+\cdots+a_{n}x_{n}^{2}$, with $a_{i}\in k$ and positive for all $i$. In fact, we can suppose that the coefficients are in $\mathcal{O}_{k}$ if we replace $e_{i}$ by $d_{i}e_{i}$ where $d_{i}\in\mathcal{O}_{k}$ is a denominator of $a_{i}$. By admissibility, $f^{\sigma}$ is positive definite for any non-trivial Galois embedding $\sigma:k\rightarrow\mathbb{R}$, hence $\sigma(a_{0})=-\sigma(-a_{0})=-f^{\sigma}(e_{0})<0$ and $\sigma(a_{i})=f^{\sigma}(e_{i})>0$ for all $i=1,\ldots,n$.

Let $E^{\prime}$ be the subspace generated by $\{e_{0},e_{1},e_{2}\}$, and $f^{\prime}:E^{\prime}\rightarrow k$ the restriction of $f$ to $E^{\prime}$. The inclusion $E^{\prime}\rightarrow E$ defines a natural inclusion $\Gamma^{\prime}=\mathrm{Spin}_{f^{\prime}}(\mathcal{O}_{k})\hookrightarrow\mathrm{Spin}_{f}(\mathcal{O}_{k})=\Gamma$. For any $\alpha\in\mathcal{O}_{k}$ and $\tau\in(\mathcal{O}_{k}\backslash\alpha\mathcal{O}_{k})^{\times}$ of order two, by definition we get an inclusion

Consider an isometric embedding of $\mathbb{H}^{2}$ into $\mathbb{H}^{n}$ equivariant with respect to the actions of $\Gamma^{\prime}$ and $\Gamma$ and the inclusions above. For any $\alpha$ and $\tau$ as before, we obtain a totally geodesic embedding

where $S_{\alpha,\tau}=\Gamma^{\prime}_{\tau}(\alpha)\backslash\mathbb{H}^{2}$ and $M_{\alpha,\tau}=\Gamma_{\tau}(\alpha)\backslash\mathbb{H}^{n}$. This implies in particular that $\mathrm{sys}(M_{\alpha,\tau})\leq\mathrm{sys}(S_{\alpha,\tau})$.

A direct computation shows that the Clifford algebra $\mathscr{C}^{+}(f^{\prime},k)$ is a quaternion algebra of $\Gamma^{\prime}$ (see [MR03, Section 12.2]). In fact, it coincides with the invariant quaternion algebra. It is well known that closed geodesics in $S^{\prime}=\Gamma^{\prime}\backslash\mathbb{H}^{2}$ are related with quadratic extensions of $k$ that embed in $\mathscr{C}^{+}(f^{\prime},k)$. In this subsection we will recall the important properties of this connection that will be useful in the sequel.

For any $s\in\mathscr{C}^{+}(f^{\prime},k)$ the reduced trace $a=s+s^{*}$, and the reduced norm $b=ss^{*}$ are elements of $k$. Hence $s$ is a root of the quadratic polynomial $g(x)=x^{2}-ax+b\in k[x]$. If $g$ is irreducible over $k$, and $L\supset k$ is the quadratic extension where $g$ splits, then for any fixed root $\alpha$ of $g$ in $L$, there exists a unique monomorphism $\phi:L\to\mathscr{C}^{+}(f^{\prime},k)$ such that $\phi(\alpha)=s$, $\phi\mid_{k}$ is the identity, and $\phi(\sigma(x))=\phi(x)^{*}$ for the non-trivial Galois automorphism $\sigma:L\rightarrow L$ of $L$ over $k$. In particular, via the identification of $L$ with $\phi(L)$, the map $\sigma$ coincides with the restriction of ^∗ to $L$.

The following proposition is a well-known fact about quaternion algebras which we recall here for reader convenience.