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The lower algebraic K-theory of Fuchsian groups. (English) Zbl 0986.19001

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 \({\mathbb H}^2\). F. T. Farrell and 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.
Reviewer: Li Fu-an (Beijing)

MSC:

19A31 \(K_0\) of group rings and orders
19D35 Negative \(K\)-theory, NK and Nil
19B28 \(K_1\) of group rings and orders

Citations:

Zbl 0798.57018
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