Small measure expansion in connected noncompact groups

Abstract

Assuming $G$ is unimodular and $K>1$, and that $\mu (A^2)<K\mu (A)$. We show that this is a continuous surjective group homomorphism $\chi :G\to L$ with a compact kernel, where $L$ is a Lie group satisfying $$\dim (L)\le \lfloor \log K\rfloor (\lfloor \log K\rfloor +1)/2.$$ This dimension bound is sharp, We also establish the relationship between $A$ and its image under the quotient map, and obtain a more general version of this result for the product set $AB$ without assuming unimodularity. We present two applications: (1) An inverse Kemperman theorem in noncompact groups. Let $G$ be a connected, noncompact, locally compact group with a left Haar measure $\mu$, and let $A$ and $B$ be nonempty, compact subsets of $G$. We classify all cases where $G$, $A$, and $B$ exhibit nearly minimal expansion properties (in the unimodular case, this means $\mu (AB)$ is close to $\mu (A)+\mu (B)$). This result completes the noncompact cases of the Kemperman inverse problem and resolves a question posed by Griesmer and Tao. (2) An effective generalized Margulis lemma for discrete subgroups of almost-connected locally compact groups acting properly by isometries. If $B(x,4r)$ can be covered by $K$ translates of $B(x,r)$ at one point and one scale, it gives a small radius $r/2M$, and the group generated at this radius has a subgroup of index at most $M\exp ((\log K{)}^4)$ which is nilpotent modulo a finite normal subgroup. The nilpotency step is bounded by $(\log K{)}^2$, and \[ M\leq K^{\exp(C(\log K)^4\log\log K)} \] and $C$ is an absolute constant.

arXiv comment33 pages, introduction and application updated

References in corpus (1)

• Minimal and nearly minimal measure expansions in connected unimodular groups

Cited by in corpus (2)

• On piecewise hyperdefinable groups
• Closed approximate subgroups: compactness, amenability and approximate lattices