Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant metric and let $m$ be left-invariant Haar measure on $G$.

Now that $G$ is a metric space its Hausdorff dimension is well defined.

1. Does the Hausdorff dimension of $G$ agree with the dimension of $G$ as a manifold?

2. Is $m$ exact dimensional? ($m$ is ``exactly $\alpha$-dimensional'' if $\lim_{r \to 0} \frac{\log m(B_r(g))}{\log r} = \alpha$ for $m$-a.e. $g$). If the answer to #1 is yes, then presumably one would take $\alpha = \dim(G)$ here.

Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$.

Now that $G$ is a metric space its Hausdorff dimension is well defined.

1. Does the Hausdorff dimension of $G$ agree with the dimension of $G$ as a manifold?

2. Is $m$ exact dimensional? ($m$ is ``exactly $\alpha$-dimensional'' if $\lim_{r \to 0} \frac{\log m(B_r(g))}{\log r} = \alpha$ for $m$-a.e. $g$). If the answer to #1 is yes, then presumably one would take $\alpha = \dim(G)$ here.