# Geometry and Geometric Analysis Working Group

#### On embedding finitely generated groups of polynomial growth and nilpotent Lie groups into uniformly convex spaces

Speaker: Seung-Yeon Ryoo, Princeton University

Location: Warren Weaver Hall 1314

It is well-known that not virtually abelian finitely generated groups of polynomial growth and nonabelian simply connected nilpotent Lie groups fail to embed bilipschitzly into uniformly convex Banach spaces. We provide quantitative bounds on this failure of embedding and a positive result on embedding into Euclidean spaces of bounded dimension. First, we prove that a ball of radius $$n$$ in the aforementioned groups incurs bilipschitz distortion $$O((\log n)^{1/q})$$ into a $$q$$-uniformly convex space; this result is optimal when considering the $$L^p$$, $$1<p<\infty$$ spaces; in particular, the $$L^2$$-distortion of such a ball is $$\Theta(\sqrt{\log n})$$. This is proven using a vector-valued Littlewood-Paley-Stein theory method inspired by Lafforgue and Naor (2012). In the Carnot group case, an alternative proof is given by proving a version of the Dorronsoro theorem. Second, we prove that for each Carnot group there is a fixed integer $$D$$ such that a ball of radius $$n$$ in the Carnot group embeds $$O(\sqrt{\log n})$$-bilipschitzly into $$\mathbb{R}^D$$; this is done by building on the approach of Tao (2018) and involves a new extension result for orthonormal vector fields. We conjecture that this embedding result can be carried out in the generality of the aforementioned groups.