New hyperbolic 4–manifolds of low volume

Abstract

We prove that there are at least two commensurability classes of (cusped, arithmetic) minimal-volume hyperbolic 4-manifolds. Moreover, by applying a well-known technique due to Gromov and Piatetski-Shapiro, we build the smallest known non-arithmetic hyperbolic 4-manifold.

Embedding arithmetic hyperbolic manifolds

“…Indeed, Theorem 1.1 applies to many of the Gromov-Piatetski-Shapiro non-arithmetic manifolds [15] (several explicit 2-and 3-dimensional examples can be constructed, c.f. [29] for a 4-dimensional one) and their generalisations [14,27,28,34], as well as to the ones introduced by Agol [1] and Belolipetsky-Thomson [7] (c.f. also [26]).…”

A small cusped hyperbolic 4‐manifold

“…(3) 𝜒 = 2 non-orientable (so also 𝜒 = 4 orientable), arithmetic [18,24]; (4) 𝜒 = 3 non-orientable and 𝜒 = 5 orientable, non-arithmetic [24].…”

Small‐volume hyperbolic manifolds constructed from Coxeter groups

“…= 120$ the group generated by the reflections in the facets of the ideal 1‐rectified Coxeter 5‐cell $$\cal R=r_1\hat{S}_{\rm reg}\subset \mathbb {H}^4$$. The manifold $$M_*^4$$ is commensurable to the orientable manifold $$N^4$$ with $$\chi (N^4)=1$$ constructed by Riolo and Slavich in a different way; see [24, 25].…”

“…= 120 the group generated by the reflections in the facets of the ideal 1-rectified Coxeter 5cell  = 𝑟 1 Ŝreg ⊂ ℍ 4 . The manifold 𝑀 4 * is commensurable to the orientable manifold 𝑁 4 with 𝜒(𝑁 4 ) = 1 constructed by Riolo and Slavich in a different way; see [24,25].…”