Poincar\'e profiles of Lie groups and a coarse geometric dichotomy

Poincar\'e profiles are a family of analytically defined coarse invariants, which can be used as obstructions to the existence of coarse embeddings between metric spaces. In this paper we calculate the Poincar\'e profiles of all connected unimodular Lie groups, Baumslag-Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. In the case of Lie groups, we obtain a dichotomy which extends both the dichotomy separating rank one and higher rank semisimple Lie groups and the dichotomy separating connected solvable unimodular Lie groups of polynomial and exponential growth. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. Our results have many consequences for coarse embeddings, for instance we deduce that for groups of the form $N\times S$, where $N$ is a connected nilpotent Lie group, and $S$ is a simple Lie group of real rank 1, both the growth exponent of $N$, and the Ahlfors-regular conformal dimension of $S$ are non-decreasing under coarse embeddings. These results are new even in the quasi-isometric setting and give obstructions to quasi-isometric embeddings which in many cases are stronger than those previously obtained by Buyalo-Schroeder.Comment: 49 pages. v2: the paper has been restructured, the main results are the same but have been presented differentl

Obstructions to embeddability into hyperquadrics and explicit examples

We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further study the embeddability problem for real-analytic submanifolds of higher codimension and answer a question by Forstneri\v{c}.Comment: Revised version, appendix and references adde

Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400 orientable manifolds, only 149 of them have hyperbolic torsion linking forms and are thus candidates for embedability in the 4-sphere. The majority of this paper is devoted to the embedding problem for these 149 manifolds. At present 41 are known to embed. Among the remaining manifolds, embeddings into homotopy 4-spheres are constructed for 4. 67 manifolds are known to not embed in the 4-sphere. This leaves 37 unresolved cases, of which only 3 are geometric manifolds i.e. having a trivial JSJ-decomposition.Comment: 58 pages, 80+ figures. V6: Included references to libraries valid in Regina 5.0+. Incorporated changes suggested by Ahmed Issa, following from his techniques developed with McCoy. Included a few recent references. To appear in Experimental Mathematic