Hyperbolic Geometry
There is also a simple geometric proof that that G = SL2(Z) is not boundedly generated. It acts by Möbius transformations on the upper half-plane H, with the Poincaré metric. Any compactly supported 1-form α on a fundamental domain of G extends uniquely to a G-invariant 1-form on H. If z is in H and γ is the geodesic from z to g(z), the function defined by
satisfies the first condition for a pseudocharacter since by the Stokes theorem
where Δ is the geodesic triangle with vertices z, g(z) and h−1(z), and geodesics triangles have area bounded by π. The homogenized function
defines a pseudocharacter, depending only on α. As is well known from the theory of dynamical systems, any orbit (gk(z)) of a hyperbolic element g has limit set consisting of two fixed points on the extended real axis; it follows that the geodesic segment from z to g(z) cuts through only finitely many translates of the fundamental domain. It is therefore easy to choose α so that fα equals one on a given hyperbolic element and vanishes on a finite set of other hyperbolic elements with distinct fixed points. Since G therefore has an infinite-dimensional space of pseudocharacters, it cannot be boundedly generated.
Dynamical properties of hyperbolic elements can similarly be used to prove that any non-elementary Gromov-hyperbolic group is not boundedly generated.
Read more about this topic: Boundedly Generated Group, Free Groups Are Not Boundedly Generated
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