Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Limits of almost homogeneous spaces and their fundamental groups

Sergio Zamora

Abstract

We say that a sequence of proper geodesic spaces $X_n$ consists of \textit{almost homogeneous spaces} if there is a sequence of discrete groups of isometries $G_n \leq \text{Iso}(X_n)$ with $\text{diam} (X_n/G_n)\to 0$ as $n \to \infty$. We show that if a sequence $(X_n,p_n)$ of pointed almost homogeneous spaces converges in the pointed Gromov--Hausdorff sense to a space $(X,p)$, then $X$ is a nilpotent locally compact group equipped with an invariant geodesic metric. Under the above hypotheses, we show that if $X$ is semi-locally-simply-connected, then it is a nilpotent Lie group equipped with an invariant sub-Finsler metric, and for $n$ large enough, $π_1(X) $ is a subgroup of a quotient of $ π_1(X_n) $.

Limits of almost homogeneous spaces and their fundamental groups

Abstract

We say that a sequence of proper geodesic spaces consists of \textit{almost homogeneous spaces} if there is a sequence of discrete groups of isometries with as . We show that if a sequence of pointed almost homogeneous spaces converges in the pointed Gromov--Hausdorff sense to a space , then is a nilpotent locally compact group equipped with an invariant geodesic metric. Under the above hypotheses, we show that if is semi-locally-simply-connected, then it is a nilpotent Lie group equipped with an invariant sub-Finsler metric, and for large enough, is a subgroup of a quotient of .

Paper Structure

This paper contains 22 sections, 56 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Lower semi-continuity of $\pi_1$
  3. Existence of the limit
  4. Further problems
  5. Summary
  6. Acknowledgments
  7. Preliminaries
  8. Notation
  9. Groups of isometries
  10. Lie groups and Hilbert's fifth problem
  11. Gromov--Hausdorff convergence
  12. Constructing covering spaces
  13. Local groups
  14. Ultralimits
  15. Ultraconvergence of polynomials
  16. ...and 7 more sections

Key Result

Theorem \oldthetheorem

(Gromov--Pansu) Let $(X,p)$ be a pointed proper geodesic space, and $G \leq \mathop{\mathrm{Iso}}\nolimits (X)$ a discrete group of isometries with $\mathop{\mathrm{diam}}\nolimits (X/G) < \infty$. If for some sequence of positive numbers $\lambda_n \to \infty$, one has in the pointed Gromov--Hausdorff sense, then $Y$ is a simply connected nilpotent Lie group equipped with a Carnot--Caratheodory

Theorems & Definitions (138)

  • Remark \oldthetheorem
  • Example \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • ...and 128 more