Hyperbolic Manifolds of Small Volume

We conjecture that for every dimension $n \neq 3$ there exists a noncompact hyperbolic $n$-manifold whose volume is smaller than the volume of any compact hyperbolic $n$-manifold. For dimensions $n \le 4$ and $n = 6$ this conjecture follows from the known results. In this paper we show that the conjecture is true for arithmetic hyperbolic $n$-manifolds of dimension $n\ge 30$.

2010 Mathematics Subject Classification: 22E40 (primary); 11E57, 20G30, 51M25 (secondary)

Keywords and Phrases: hyperbolic manifold, volume, Euler characteristic, arithmetic group.

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