The problem
Question (Gromov): is every Gromov-hyperbolic group residually finite? That is, for every non-identity \(g\) does there exist a homomorphism to a finite group not killing \(g\)?
Residual Finiteness of Hyperbolic Groups
openHistory & significance
Gromov's 1987 essay founding hyperbolic group theory asked which of their properties mirror 3-manifold groups — residual finiteness among the most basic. Agol–Wise virtual specialness (2012) settled virtual Haken-type questions for many hyperbolic groups without implying residual finiteness; non-residually-finite hyperbolic examples would have to be non-linear (all linear groups are residually finite by Malcev), and none are known. A counterexample would be as shocking as a proof.
Connected problems
- related Existence of Non-Sofic Groups
More in geometric group theory
- same area The Hanna Neumann conjecture
- same area Cannon's conjecture
- same area Amenability of Thompson's group F
References
Still open.
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