# Maximally symmetric hyperbolic 3-manifolds with finite volume

In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $$S^3$$ or less popular the hyperbolic space $$H^3$$). Since $$S^3$$ has a finite volume it seems natural to ask whether there also exists a maximally symmetric hyperbolic counterpart which also has a finite volume? (An answer in layman's terms would be fine, if possible.)

• You probably want to read about Thurston geometries. Also about the notion of "locally symmetric space". That $H^3$ is "less popular"? well, hyperbolic 3-dimensional geometry has been a major theme in geometric topology in the last 40 years! – YCor May 20 at 16:53
• Sorry, but less popular in standard cosmology, because it is generally believed that space-time in general relativity is not negatively curved. – layman May 20 at 16:58
• In any case: finite volume manifolds with curvature $-1$ usually have a very small amount of symmetries, whence the notion of locally symmetric space. – YCor May 20 at 17:17
• In GR we sometimes say "locally maximally symmetric" to mean: for every point on the manifold there exists an open neighbourhood of said point which is isometrically diffeomorphic to an open set on a "maximally symmetric" space. So for the purposes of this question said maximally symmetric space would be $H^3$. There should be lots of examples* which are not $H^3$ itself, but whether any have finite volume, I do not know. (*In the Lorentzian case the BTZ black hole geometry is some quotient of maximally symmetric $AdS_3$ but I do not remember the details now.) – AlexArvanitakis May 20 at 19:37
• Equivalently, such locally maximally symmetric spaces admit the maximum number of Killing vector fields on some neighbourhood of every point. – AlexArvanitakis May 20 at 19:39