The object of this paper, which is the first in a series of three, is to lay the foundations of the theory of ideals and algebraic sets over groups.

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A group G is called a triangle group if it can be presented in the formIt is well-known that G is isomorphic to a subgroup of PSL2(ℂ), that a is of order l, b is of order m and ab is of order n. Ifthen G contains the fundamental group of a positive genus orientable surface as a subgroup of finite index whenever s(G) ≤ 1; in particular G is infinite. Furthermore, if s(G) < 1, the genus of the surface is greater than 1 and consequently G contains a free group of rank 2.

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