Matrix Representation of the Isometry Group

Abstract

It is shown that the group of direct isometries of the hyperbolic plane is isomorphic to the group of fractional linear transformations with real coefficients. The coefficients of such a transformation can be taken as the entries of a 2 × 2 real matrix with determinant 1; there is a one-to-two relation between the transformations and such matrices, where a matrix M and its negative −M lead to the same fractional linear transformation, hence to the same isometry of the hyperbolic plane. Those matrices constitute the group SL(2, ℝ) of 2 × 2 real matrices of determinant 1, and it follows that the group of direct isometries of the hyperbolic plane is isomorphic to the quotient group of SL(2, ℝ) with respect to its subgroup consisting of I and −I,where I is the 2 × 2 unit matrix. This makes it possible to study the isometries by studying the matrices. The points, lines, and curves that are invariant (mapped onto themselves) under an isometry are classified. It is proved that the direct isometry group of the hyperbolic plane is simple, has no nontrivial proper normal subgroups, hence differs in this respect from the isometry group of the Euclidean plane, where the translations form a normal subgroup. Lastly, the subgroup SL(2, ℤ) of SL(2, ℝ) consisting of matrices with integer entries is studied; it leads to a tiling of the hyperbolic plane by triangles each having two 60° vertices and one 0° vertex at infinity.