Quasi-arithmeticity of lattices in $${{\mathrm{PO}}}(n,1)$$ PO ( n , 1 )

Abstract

Abstract. We show that the non-arithmetic lattices in PO(n, 1) of Belolipetsky and Thomson [BT11], obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and PiatetskiShapiro are not quasi-arithmetic. A corollary of this is that there are, for all n 2, non-arithmetic lattices in PO(n, 1) that are not commensurable with the Gromov-Piatetski-Shapiro lattices.

Nonarithmetic hyperbolic manifolds and trace rings

“…It is an easy consequence of the lemma that all doubly-cut glueings are quasi-arithmetic. This fact is also proven in [Tho16].…”

“…Manifolds obtained via this construction will be referred to as doubly-cut glueings and the two corresponding hyperplanes as the cut hyperplanes (see Section 1.2).An interesting consequence is that infinitely many of such doubly-cut glueings are nonarithmetic and pairwise non-commensurable. Moreover, these are the first examples in arbitrary dimension of nonarithmetic manifolds that are quasi-arithmetic (see [Tho16]). However if one is only interested in constructing nonarithmetic manifolds, their proof is somehow nonexplicit in the sense that it relies on the systole argument for proving both nonarithmeticity and pairwise non-commensurability.…”

On volumes of quasi-arithmetic hyperbolic lattices

“…Then it follows from the work of Vinberg [27] that Γ ′ ⊂ G(k) (using the fact that G is adjoint). In particular, this shows that our definition of quasi-arithmeticity coincides with the one from [26] and [24]. Remark 1.2.…”

“…First examples were obtained by Vinberg, who considered reflection groups [26]. The construction of Belolipetsky and Thomson [2] proves the existence of infinitely many commensurability classes of properly quasi-arithmetic hyperbolic lattices in any dimension n > 2; see Thomson [24]. Remark 1.1.…”

Embedding arithmetic hyperbolic manifolds

“…In the latter case M is always "quasi-arithmetic" (c.f. [26,32,33] for this notion), in contrast to the former case [32]. In both cases, there are infinitely many commensurability classes of such manifolds [28,32], and thus we have:…”