On a compact Lie group with a left-invariant Riemannian metric, many important functional inequalities for the Laplacian (such as Poincar\'e inequality, parabolic Harnack inequality, etc.) can be proved using only the volume doubling property. That is, constants in these inequalities can be controlled by the doubling constant of the metric; this can be strictly more powerful than classical techniques involving Ricci curvature lower bounds. It can happen that there is a uniform bound on the doubling constants of all left-invariant metrics on a given Lie group; such a group is called uniformly doubling. In such a case, the implicit constants in the functional inequalities will also be uniformly bounded over all left-invariant metrics. In the joint work with Nate Eldredge and Laurent Saloff-Coste we showed that the special unitary group SU(2) is iniformly doubling via explicit uniform volume estimates and described the consequences (heat kernel estimates, Weyl counting function etc). I will report on the recent progress on related results on SU(2)x\mathbb{R}^{n} and the MCP on SU(2) (joint with C. Rigoni).