Abstract
We describe new topological invariants of the classical action of SL 2(ℤ[1/m]) on the hyperbolic plane, where m is a positive integer. One defines the notion of this action (or any action of a discrete group on a CAT(0) metric space) as being 'controlled s-connected' where s is an integer ≥−1. In the case of this classical action, we show it is (s − 2)-connected but not (s − 1)-connected where s is the number of different primes dividing m. It turns out that the behavior of this action is quite different over rational and irrational endpoints, with the bounds on connectivity occurring only at the rational end points. The very lowest case, m=1, reduces to elementary and seemingly innocuous statements about how SL 2(ℤ) acts on the hyperbolic plane. Extensions involving other Fuchsian groups are also given.
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