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We relate the geometrical and the Chern-Simons description of(2+1)-dimensional gravity for spacetimes of topology $R\times S_g$, where $S_g$is an oriented two-surface of genus $g>1$, for Lorentzian signature and generalcosmological constant and the Euclidean case with negative cosmologicalconstant. We show how the variables parametrising the phase space in theChern-Simons formalism are obtained from the geometrical description and howthe geometrical construction of (2+1)-spacetimes via grafting along closed,simple geodesics gives rise to transformations on the phase space. Wedemonstrate that these transformations are generated via the Poisson bracket byone of the two canonical Wilson loop observables associated to the geodesic,while the other acts as the Hamiltonian for infinitesimal Dehn twists. Forspacetimes with Lorentzian signature, we discuss the role of the cosmologicalconstant as a deformation parameter in the geometrical and the Chern-Simonsformulation of the theory. In particular, we show that the Lie algebras of theChern-Simons gauge groups can be identified with the (2+1)-dimensional Lorentzalgebra over a commutative ring, characterised by a formal parameter$\Theta_\Lambda$ whose square is minus the cosmological constant. In thisframework, the Wilson loop observables that generate grafting and Dehn twistsare obtained as the real and the $\Theta_\Lambda$-component of a Wilson loopobservable with values in the ring, and the grafting transformations can beviewed as infinitesimal Dehn twists with the parameter $\Theta_\Lambda$.Comment: 50 pages, 6 eps figure

Geometrical (2+1)-Gravity and the Chern-Simons Formulation: Grafting, Dehn Twists, Wilson Loop Observables and the Cosmological Constant