On the isoperimetric profile in infinite dimensional pro-Lie groups

Abstract

Let $K$ be a nontrivial compact connected Lie group with a bi-invariant Riemannian metric. On the compact pro-Lie group $K^{\N}$ we define a weighted variation by summing the distributional variations in its Lie group coordinates. We prove that all sublevel sets with bounded $L^1$-norm and weighted variation are compact in $L^1$ if and only if the coordinate weights tend to infinity. The proof gives a quantitative estimate for approximation by Haar averages over normal subgroups, illustrating that it is possible to apply a methodology which is typical of the geometric measure theory in a specific area of the topological group theory. The resulting perimeter is invariant under left and right translations and is preserved under pullback by quotient homomorphisms equipped with quotient metrics. Isoperimetric minimizers exist at every prescribed Haar probability. For the infinite torus, the compactness condition is equivalent to properness of a weighted length on its character group and to compactness of a Fourier multiplier's resolvent. Finally, we show that locally compact noncompact groups require an additional condition of tightness, in order to be treated as an appropriate generalization of the compact case.