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The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter group; the usual notion of simple continued fractions comes from adding such a generator to $\tilde{I}_1$.

Given a regular tiling of a two-dimensional space (whether spherical, Euclidean, or hyperbolic), we get a triangle group by barycentrically subdividing the regular polygons and considering the ways of reflecting the triangles into each other. The triangle group acts on the space in the obvious way.

Say we pick the center of one of the original polygons as our origin, pick units so that the hypotenuse of the triangle has length 1, and pick one of the triangles with a vertex at the origin as the canonical one.

We can act on the origin with an element of the triangle group and reflect it outside the unit circle, then geometrically invert the point and bring it back inside. Those points on the surface that one can reach in a finite number of such steps could be thought of as "rational".

In the Euclidean and hyperbolic cases, we can also go the other way, since geometric inversion always takes a point inside the unit circle to a point outside of it. We can act on a point in the space with a group element and reflect it into the canonical triangle, then do geometric inversion to place it outside the unit circle. Those points that reach the origin in a finite number of moves could be called "rational"; those points with repeating continued fractions could be called "quadratic surds".

Does anyone know of previous work on this idea? Does the generalization lead to any interesting number theory?

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1 Answer 1

The matrix representation of continued fractions appears in Milne-Thomson, "The Calculus of Finite Differences", Chelsea, 1981. As far as I know he was the first to study them systematically.

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Whatever the virtues of the cited reference, it is not true that 1981 was the first year "matrix representations of" continued fractions were "studied systematically", given prior ambient awareness of their obvious properties. E.g., in 1975 Nick Katz remarked in the common room at Princeton that it was "obvious" that the theory of continued fractions is a study of the action of $SL_2(\mathbb Z)$, especially, tracking its generators sending $z$ to $z+1$ and/or to $-1/z$, on the real line. Various attempts to "generalize" the ideas were extant in those years... not very interesting, I recall. –  paul garrett Nov 26 '12 at 1:06
Paul: 1981 is only the year that Chelsea reprinted the book. Milne-Thomson worked in the early 20th century and that book of his is from the 1930s. –  KConrad Nov 26 '12 at 6:55
Aha! @KConrad, thanks for the info! (Sorry to be slow to notice...) –  paul garrett Dec 7 '12 at 18:32

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