What is known about the ($L^2$) spectrum of the minus Laplace-Beltrami operator ($- \Delta$) with zero boundary conditions on $B =H^n/\Gamma$, where $H^n$ is $n$-dimensional hyperbolic space ($n>1$), $\Gamma$ is a discrete subgroup of the isometry group of $H^n$? Is the spectrum discrete when $B$ has finite volume? If so, is it discrete for generic $B$ (with smooth boundary) of finite volume? (When the closure of $B$ is compact the spectrum seems to be discrete.) Are there universal bounds (from below) on the spectrum of ($- \Delta$) on $B$ of finite volume (with smooth boundary)? The case of special interest is $n=2$ and $\Gamma = PSL(2,{\mathbb Z})$ - well-known modular group.

What is known about the ($L^2$) spectrum of the minus Laplace-Beltrami operator ($- \Delta$) with zero boundary conditions on $B =H^n/\Gamma$, where $H^n$ is $n$-dimensional hyperbolic space ($n>1$), $\Gamma$ is a discrete subgroup of the isometry group of $H^n$? Is it possible for the spectrum to be discrete for some $\Gamma$, when $B$ has finite volume but is not (sub)compact? If so, it could be generalized for a large class of (generic) $B$ (with smooth boundary) of finite volume. (When the closure of $B$ is compact the spectrum seems to be discrete.) Are there universal bounds (from below) on the spectrum of ($- \Delta$) on $B$ of finite volume (with smooth boundary)? The case of special interest is $n=2$ and $\Gamma = PSL(2,{\mathbb Z})$ - well-known modular group.