# survey paper on the construction of hyperbolic manifolds

Is there a good survey paper which discusses the common ways of building hyperbolic $$n$$-manifolds?

• Section 3.1 here. – Misha Jan 5 at 13:15
• See Dave Witte-Morris' book for arithmetic groups, and section 6.5 for the non-arithmetic Gromov-Piatetskii-Shapiro examples. deductivepress.ca These and variants (together with Selberg's Lemma) are essentially the only known general method for proving the existence of finite volume hyperbolic $n$-manifolds (variants are given here arxiv.org/abs/1802.04619). In 3 dimensions, in some sense the geometrization theorem gives a construction of all finite volume hyperbolic 3-manifolds. The Deligne-Mostow construction gives some examples as moduli spaces of polygons. – Ian Agol Jan 5 at 18:19
• For recent constructions of finite volume hyperbolic manifolds with various properties, see: arxiv.org/abs/1008.2646 arxiv.org/abs/1507.02747 arxiv.org/abs/1703.10561 arxiv.org/abs/1812.06536 arxiv.org/abs/1904.12720 – Ian Agol Jan 5 at 18:21