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A symmetry based quantization method of reparametrization invariant systemsis described; it will work for all systems that possess complete sets ofperennials whose Lie algebras close and which generate a sufficiently largesymmetry groups. The construction leads to a quantum theory including a Hilbertspace, a complete system of operator observables and a unitary time evolution.The method is applied to the 2+1 gravity. The paper is restricted to themetric-torus sector, zero cosmological constant $\Lambda$ and it makes stronguse of the so-called homogeneous gauge; the chosen algebra of perennials isthat due to Martin. Two frequent problems are tackled. First, the Lie algebraof perennials does not generate a group of symmetries. The notion of groupcompletion of a reparametrization invariant system is introduced so that thegroup does act; the group completion of the physical phase space of our modelis shown to add only some limit points to it so that the ranges of observablesare not unduly changed. Second, a relatively large number of relations betweenobservables exists; they are transferred to the quantum theory by thewell-known methods due to Kostant and Kirillov. In this way, a uniqueness ofthe physical representation of some extension of Martin's algebra is shown. TheHamiltonian is defined by a systematic procedure due to Dirac; for the torussector, the result coincides with that by Moncrief. The construction may beextensible to higher genera and non-zero $\Lambda$ of the 2+1 gravity, becausesome complete sets of perennials are well-known and there are no obstructionsto the closure of the algebra.Comment: The first two sections completely reworked to make the paper clearer  and more self-contained. Some misprints corrected, and some insights added in  the main bod

Group-theoretical quantization of 2+1 gravity in the metric-torus sector