2021/05/14 by Mikhail Belolipetsky, Belolipetsky, Mikhail, Nikolay Bogachev +5 · 3 citations
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic geometry #Arithmetic #Geodesic #Geometric and Algebraic Topology #Hyperbolic function #Hyperbolic geometry #Hyperbolic manifold #Hyperbolic space #Hyperbolic triangle #Linear subspace #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Orbifold #Pure mathematics #Relatively hyperbolic group #Subspace topology
paper · pdf · doi:10.56994/jamr.004.001.003
published in Journal of the Association for Mathematical Research. 4(1), 111-182
openalex created_date 2021/05/24 · openalex publication_date 2026/03/12 · openalex updated_date 2026/08/05
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 ⩾ ⌊n/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, exhibiting 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 commensu rability 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.