[Paper Review] Separable and tree-like asymptotic cones of groups

Alessandro Sisto|arXiv (Cornell University)|Oct 6, 2010

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper uses nonstandard analysis to characterize which metric spaces can arise as asymptotic cones of groups, proving that groups with few separable asymptotic cones are virtually nilpotent and classifying the real trees that can appear as asymptotic cones—specifically, only points, lines, or trees with $2^{\aleph_0}$-valence at each point.

ABSTRACT

Using methods from nonstandard analysis, we will discuss which metric spaces can be realized as asymptotic cones. Applying the results we will find in the context of groups, we will prove that a group with "a few" separable asymptotic cones is virtually nilpotent, and we will classify the real trees appearing as asymptotic cones of (not necessarily hyperbolic) groups.

Motivation & Objective

• To determine which metric spaces can be realized as asymptotic cones of groups using nonstandard analysis.
• To investigate the geometric and topological constraints on asymptotic cones, particularly separability and dimension.
• To classify real trees that can appear as asymptotic cones of arbitrary (not necessarily hyperbolic) groups.
• To establish conditions under which a group must be virtually nilpotent based on the structure of its asymptotic cones.
• To show that every proper metric space can be realized as an asymptotic cone of some metric space, providing a completeness result for the construction.

Proposed method

• Employing nonstandard extensions of metric spaces and groups to define asymptotic cones via infinitesimal scaling and ultraproducts.
• Using the dichotomy that internal sets in nonstandard extensions are either finite or have cardinality at least $2^{\aleph_0}$ to derive topological constraints.
• Applying the nonstandard characterization of compactness (via internal points) to prove surjectivity in cone constructions.
• Leveraging the fact that proper asymptotic cones with finite Minkowski dimension imply virtual nilpotency, as established in prior work.
• Constructing a metric space $Y$ from a given proper metric space $X$ such that $X$ is isometric to an asymptotic cone of $Y$, using a warped product-like distance function.
• Using homogeneity and scaling invariance to analyze the structure of asymptotic cones under varying infinitesimal scaling factors.

Experimental results

Research questions

• RQ1 Which metric spaces can appear as asymptotic cones of a group or metric space?
• RQ2 Under what conditions does a group with separable asymptotic cones become virtually nilpotent?
• RQ3 What real trees can arise as asymptotic cones of non-hyperbolic groups?
• RQ4 Can every proper metric space be realized as an asymptotic cone of some other metric space?
• RQ5 How do the topological and geometric properties of asymptotic cones (e.g., separability, dimension) reflect on the original group?

Key findings

• A separable metric space can only be an asymptotic cone if it is proper, so the separable Hilbert space cannot appear as an asymptotic cone.
• If all asymptotic cones of a group $G$ with scaling factors in an interval $[\nu_1, \nu_2]$ with $\nu_1 \ll \nu_2$ are separable, then $G$ is virtually nilpotent.
• The only real trees that can appear as asymptotic cones of a group are points, the real line, or trees in which every point has valency $2^{\aleph_0}$.
• Every proper metric space $X$ is isometric to an asymptotic cone of some metric space $Y$, and if $X$ is geodesic and unbounded, $Y$ can be chosen to be geodesic as well.
• The construction of $Y$ from $X$ uses a nonstandard extension of $X \times \mathbb{N}$ with a carefully defined distance function to ensure the asymptotic cone recovers $X$.
• The proof relies on the nonstandard characterization of compactness: an internal point in a nonstandard extension of a compact set is infinitesimally close to a standard point.