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

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    $\begingroup$ Section 3.1 here. $\endgroup$
    – Misha
    Commented Jan 5, 2020 at 13:15
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    $\begingroup$ 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. $\endgroup$
    – Ian Agol
    Commented Jan 5, 2020 at 18:19
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    $\begingroup$ 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 $\endgroup$
    – Ian Agol
    Commented Jan 5, 2020 at 18:21

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Not a paper, but a book : Foundations of Hyperbolic Manifolds by John Ratcliffe. Chapters 10 and 11 might contain what you're looking for. Also available here.

Concerning 4-manifolds, you have a survey by Bruno Martelli here.

Also, I'm sure you can find what you want typing construction of hyperbolic manifolds on your favorite search engine.

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