The lower algebraic K-theory of Fuchsian groups (Q5943653)

A Fuchsian group is a discrete subgroup of \(\text{PSL}_2 ({\mathbb R})\) or a conjugate of such a group in \(\text{PSL}_2 ({\mathbb C})\). Since \(\text{PSL}_2 ({\mathbb R})\) can be identified with the orientation-preserving isometries of the hyperbolic plane \({\mathbb H}^2\), the Fuchsian groups act in a natural way on \({\mathbb H}^2\). \textit{F. T. Farrell} and \textit{L. E. Jones} [J. Am. Math. Soc. 6, No. 2, 249-297 (1993; Zbl 0798.57018)] proved that there is a significant simplification in calculating the lower algebraic \(K\)-theory of discrete cocompact subgroups of virtually connected Lie groups and their subgroups. In this paper, the authors show that a further reduction can be made, and use spectral sequences to calculate the lower algebraic \(K\)-groups of the integral group ring \({\mathbb Z} \Gamma\), where \(\Gamma\) is a cocompact Fuchsian group.