On the sphericity of the subgroup \(\mathrm{PSL}_2(\mathbb{R})\) in the group of diffeomorphisms of the circle (Q317286)

Let \(\mathrm{SL}_2 (\mathbb{R})\) be the group of \(2\times2\) real matrices with determinant 1, \(\mathrm{PSL}_2 (\mathbb{R})\) be its quotient with respect to the center, and \(\mathrm{SL}_2 (\mathbb{R})^{\sim}\) be its universal covering group. The main result is the following theorem. a) The subgroup \(\mathrm{PSL}_2 (\mathbb{R})\) is spherical in the group of \(C^3\) orientation-preserving diffeomorphisms of the circle (denoted by \(\mathrm{Diff}^3\)). b) The subgroup \(\mathrm{PSL}_{2} ^{\sim} (\mathbb{R})\) is spherical in the group \(\widetilde{\mathrm{Diff}}^3\) which is the central extension of \(\mathrm{Diff}^3\) determined by the Bott cocycle.