Subspace stabilisers in hyperbolic lattices

This paper shows that immersed totally geodesic $m$-dimensional suborbifolds of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite subgroups of the commensurator whenever $m \geqslant \lfloor \frac{n}{2} \rfloor$. 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 provide examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. We provide an algebraic characterisation 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.