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A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an analagous question is open already in dimension $4$ -- i.e. it is not known that $M^4$ with negative sectional curvature always have a cover of finite degree? Are there any positive results in this direction?

This a folow up to question that turned up to be open Existence of finite index torsion-free subgroups of hyperbolic groupsExistence of finite index torsion-free subgroups of hyperbolic groups

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an analagous question is open already in dimension $4$ -- i.e. it is not known that $M^4$ with negative sectional curvature always have a cover of finite degree? Are there any positive results in this direction?

This a folow up to question that turned up to be open Existence of finite index torsion-free subgroups of hyperbolic groups

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an analagous question is open already in dimension $4$ -- i.e. it is not known that $M^4$ with negative sectional curvature always have a cover of finite degree? Are there any positive results in this direction?

This a folow up to question that turned up to be open Existence of finite index torsion-free subgroups of hyperbolic groups

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Dmitri Panov
Dmitri Panov
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Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree?

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an analagous question is open already in dimension $4$ -- i.e. it is not known that $M^4$ with negative sectional curvature always have a cover of finite degree? Are there any positive results in this direction?

This a folow up to question that turned up to be open Existence of finite index torsion-free subgroups of hyperbolic groups