Subspace stabilisers in hyperbolic lattices

Abstract

This paper shows that all immersed totally geodesic suborbifolds of arithmetic hyperbolic orbifolds correspond to finite order subgroups of the commensurator. We call such totally geodesic suborbifolds fc-subspaces (finite centraliser subspaces) and use them to produce 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 nonarithmetic orbifolds that contain non-fc subspaces. Finally, we study arithmetic properties of orbifolds that descend to their totally geodesic suborbifolds and deduce some information about their geodesic embeddings. It follows that certain arithmetic hyperbolic orbifolds cannot be geodesically immersed in any other arithmetic ones.

From Geometry to Arithmetic of Compact Hyperbolic Coxeter Polytopes

“…Emery [15] recently demonstrated that the covolume of any quasi-arithmetic lattice is a rational multiple of the covolume of an arithmetic lattice associated to the same ambient group. Besides this, it was shown by Belolipetsky et al [4,Theorem 1.7] that (quasi-)arithmeticity is inherited by totally geodesic suborbifolds; more precisely, if M is a quasi-arithmetic hyperbolic orbifold with adjoint trace field k(M ), and N ⊂ M is a finite volume totally geodesic suborbifold of dimension m 2 with adjoint trace field k(N ), then N is hyperbolic and quasi-arithmetic, with k(M ) ⊆ k(N ).…”

“…For quasi-arithmetic straight Coxeter prisms this implies, in particular, that their triangle base (which is orthogonal to its neighbors) should also be quasiarithmetic, and even arithmetic according to Remark 2.4. In fact, one could even apply even a more recent result in this situation: the above mentioned triangle base F is a totally geodesic reflective suborbifold of an ambient prismatic Coxeter orbifold P , and by [4,Theorem 1.7], if P is quasi-arithmetic, then F should be quasi-arithmetic as well.…”

“…For type-8 prisms, we consider the following non-trivial cyclic product cos π 4 cos π k cos π m : if k = 3 then m = 4, since otherwise, this cyclic product belongs to some extension of Q, and if k = 4 then only m = 3 or m = 6 are possible. Thus, for type 8 only the following options are available to give a quasi-arithmetic reflection group: (k, 4, m) = (3, 4, 4), (4,4,3), (4,4,6). Similar considerations for type 9 leave only m = 6 to be checked.…”

“…Let us now consider the more general case when a Coxeter prism is glued from two straight prisms along a common base. Recent results of Belolipetsky et al [4] allow us to rule out quasi-arithmeticity of the significantly many such Coxeter prisms by considering the adjoint trace field of totally geodesic subspaces. One can observe that in H 3 there are only three infinite families of compact Coxeter prisms (see Table 1), which we will denote by P j k,l,m where j = 1, 2, 3 denotes the type (so j corresponds to the j th diagram in the list) of the prism, and k, l, m are the diagram parameters.…”

Lannér diagrams and combinatorial properties of compact hyperbolic Coxeter polytopes

“…Generally speaking, there are two different approaches to both problems: classification of finite-volume Coxeter polytopes of some certain combinatorial types (see [Kap74, Ess96, Tum07, FT08, FT09, JT18, MZ22, Bur22]) and the theory of arithmetic hyperbolic reflection groups (see [Vin72,Bel16,Bog17,BP18,Bog19,Bog20]). In particular, in the context of arithmetic and quasi-arithmetic reflection groups several authors constructed novel Coxeter polytopes as faces or reflection centralizers of some higher dimensional polytopes (see [Bor87,All06,All13,BK21,BBKS21]).…”