I encountered this notation in a paper by Carron:

When X = $\Gamma{\setminus}\mathbb{H}^n$ is a real hyperbolic manifold, ...

$\Gamma$ is a discrete torsion free subgroup of SO$(n,1)$. My question is narrow: what is the meaning of the backslash notation? It is specifically used without spaces, i.e., not $\Gamma \setminus \mathbb{H}^n$. Thanks for clarifying!

  • $\begingroup$ Which paper? :) $\endgroup$ – J. M. isn't a mathematician Nov 30 '11 at 13:45
  • $\begingroup$ @J.M.: Gilles Carron, "Rigidity and $L^2$ cohomology of hyperbolic manifolds," Ann. Inst. Fourier, 60(7) 2010. $\endgroup$ – Joseph O'Rourke Nov 30 '11 at 13:48

This notation means that $\Gamma$ acts on $\mathbb{H}^n$ and $X$ is the quotient space for this action, i.e, the space of orbits.

Since $\Gamma \subset SO(n,1)$, the action is by isometries. Hence $X$ inherits a metric such that the canonical projection $\mathbb{H}^n \to X$ is a local isometry and therefore $X$ has constant negative curvature (justifying the name "real hyperbolic manifold").

The notation $\Gamma \backslash \mathbb{H}^n$ is justified by the fact that $\Gamma$ acts on $\mathbb{H}^n$ on the left. This is a standard notation for the orbit space of an action: if G acts on Y on the left the space of orbits is $G \backslash Y$ and if it acts on the right the notation is $Y/G$.

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  • $\begingroup$ Thanks so much, Lucas! I should have intuited the "acting on the left" aspect. $\endgroup$ – Joseph O'Rourke Nov 30 '11 at 14:06

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