Quantum holonomies in (2+1)-dimensional gravity

We describe an approach to the quantisation of (2+1)-dimensional gravity with topology image and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q-commutation relation. Solutions of diagonal and upper-triangular form are constructed, which in the latter case exhibit additional, non-trivial internal relations for each holonomy matrix. Representations are constructed and a group of transformations – a quasi-modular group – which preserves this structure, is presented.