This work provides an extension of parts of the classical finite dimensional sub-elliptic theory in the context of infinite dimensional compact connected metrizable groups. Given a well understood and well behaved bi-invariant Laplacian, $Δ$, and a sub-Laplacian, $L$, to which intrinsic distances, $d_Δ$, $d_L$, are naturally attached, we show that a comparison inequality of the form $d_L\le C(d_Δ)^c$ (for some $0<c\le 1$) implies that the Dirichlet form of a fractional power of $Δ$ is dominated by the Dirichlet form associated with $L$. We use this result to show that, under additional assumptions, certain good properties of the heat kernel for $Δ$ are then passed to the heat kernel associated with $L$. Explicit examples on the infinite product of copies of $SU(2)$ are discussed to illustrate these results.