On the first Betti numbers of hyperbolic surfaces (Q1179298)

Let \(G\) be \(SO(n,1)\) or \(SU(n,1)\) and \(\Gamma\) a cocompact, torsion-free, discrete subgroup of \(G\). Using the Selberg trace formula the author gets an upper bound of the multiplicity of the spherical complementary series \(\pi_{1,-i\alpha\rho}\), \(0<\alpha\leq 1\), of \(G\) in \(L^ 2(\Gamma\setminus G)\). From that upper bound the author estimates the first Betti numbers for \(G=SU(2,1)\).