Subspace stabilisers in hyperbolic lattices

TL;DR

This work introduces fc-subspaces, defined as fixed loci of finite subgroups of the commensurator of a lattice in $ ext{Isom}(\,\mathbb{H}^n)$, and proves an arithmeticity criterion: a finite-volume hyperbolic orbifold is arithmetic iff it contains infinitely many fc-subspaces; in contrast, non-arithmetic orbifolds have only finitely many fc-subspaces, with a bound proportional to the volume. It develops Vinberg invariants (adjoint trace field $k$ and ambient group) to analyze how fc-subspaces inherit or enlarge arithmetic data, revealing cases where the adjoint trace field of a subspace properly contains that of the ambient orbifold. The paper identifies two robust construction mechanisms for fc-subspaces—subform subspaces and Weil restriction subspaces—and uses them to classify totally geodesic immersions between non-exceptional arithmetic orbifolds; it also studies exceptional trialitarian lattices in dimension $7$, showing the existence of 3D type III fc-subspaces within such ambient spaces. Moreover, it shows that suborbifolds of (quasi-)arithmetic orbifolds remain (quasi-)arithmetic, and provides explicit examples and obstructions illustrating the richness and boundaries of fc-subspace phenomena across dimensions and lattice types.

Abstract

This paper shows that immersed totally geodesic $m$-dimensional suborbifolds of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite subgroups of the commensurator whenever $m \geqslant \frac{n-1}{2}$. We call such totally geodesic suborbifolds finite centraliser subspaces (or fc-subspaces) and use them to formulate an arithmeticity criterion for hyperbolic lattices. We show that a hyperbolic orbifold $M$ is arithmetic if and only if it has infinitely many fc-subspaces, and exhibit examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. We provide an algebraic characterization of totally geodesically immersed suborbifolds of arithmetic hyperbolic orbifolds by analysing Vinberg's commensurability invariants. This allows us to construct examples with the property that the adjoint trace field of the geodesic suborbifold properly contains the adjoint trace field of the orbifold. The case of special interest is that of exceptional trialitarian $7$-dimensional orbifolds. We show that every such orbifold contains a totally geodesic arithmetic hyperbolic $3$-orbifold of exceptional type. Finally, we study arithmetic properties of orbifolds that descend to their totally geodesic suborbifolds, proving that all suborbifolds in a (quasi-)arithmetic orbifold are (quasi-)arithmetic.

Figures (4)

• Figure 1: The Tits index of ${^6}D_{4,0}$. The reader should notice that there are no circled roots as the group is totally anisotropic. The action of the absolute Galois group $\mathcal{G} =\mathrm{Gal}(\overline{k}/k)$ is induced by complex conjugation $\sigma$, which exchanges $\alpha_2$ and $\alpha_3$, and the order $3$ trialitarian automorphism $\tau$ which permutes cyclically $\alpha_1,\, \alpha_2$ and $\alpha_3$.
• Figure 2: The Tits index (relative to the torus $\mathbf{T}$) of the centraliser $\mathbf{H}$ of the involution $\theta$. The arrows indicate the action on the Dynkin diagram of complex conjugation $\sigma$ and of the trialitarian automorphism $\tau \in \mathrm{Gal}(E/k)$.
• Figure 3: A twist knot in the $3$-sphere, together with the immersed thrice-punctured sphere $S$ (shaded). The self-intersection of $S$ is drawn as a dashed line.
• Figure 4: The Coxeter diagram of a non-compact hyperbolic $5$-simplex, with labels for its facets.

Theorems & Definitions (86)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10