Quantum holonomies in (2+1)-dimensional gravity

We describe an approach to the quantisation of (2 + 1)-dimensional gravity with topology R X T-2 and negative cosmological constant, which uses two qu antum holonomy matrices satisfying a q-commutation relation. Solutions of d iagonal and upper-triangular form are constructed, which in the latter case exhibit additional, non-trivial internal relations for each holonomy matri x. Representations are constructed and a group of transformations - a quasi -modular group - which preserves this structure, is presented. (C) 2000 Els evier Science B.V. All rights reserved.