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Geometric invariants of locally compact groups: the homotopical perspective

Kai-Uwe Bux, Elisa Hartmann, José Pedro Quintanilha

Abstract

We extend the classical theory of homotopical $Σ$-sets $Σ^n$ developed by Bieri, Neumann, Renz and Strebel for abstract groups, to $Σ$-sets $Σ_{\mathrm{top}}^n$ for locally compact Hausdorff groups. Given such a group $G$, our $Σ_{\mathrm{top}}^n(G)$ are sets of continuous homomorphisms $G \to \mathbb{R}$ ("characters"). They match the classical $Σ$-sets $Σ^n(G)$ if $G$ is discrete, and refine the homotopical compactness properties $\mathrm C_n$ of Abels and Tiemeyer. Moreover, our theory recovers the definition of $Σ_{\mathrm{top}}^1$ and $Σ_{\mathrm{top}}^2$ proposed by Kochloukova. Besides presenting various characterizations of $Σ_{\mathrm{top}}^n$ (particularly for $n\in \{1,2\}$), we show that characters in $Σ_{\mathrm{top}}^n(G)$ are also in $Σ_{\mathrm{top}}^n(H)$ if $H\le G$ is a closed cocompact subgroup, and we generalize several classical results. Namely, we prove that the set of nonzero elements of $Σ_{\mathrm{top}}^n(G)$ is open, we prove that characters in a group of type $\mathrm C_n$ that do not vanish on the center always lie in $Σ_{\mathrm{top}}^n(G)$, and we relate the $Σ$-sets of a group with those of its quotients by closed subgroups of type $\mathrm C_n$. Lastly, we describe how $Σ_{\mathrm{top}}^n(G)$ governs whether a closed normal subgroup with abelian quotient is of type $\mathrm C_n$, generalizing one of the highlights of the classical theory.

Geometric invariants of locally compact groups: the homotopical perspective

Abstract

We extend the classical theory of homotopical -sets developed by Bieri, Neumann, Renz and Strebel for abstract groups, to -sets for locally compact Hausdorff groups. Given such a group , our are sets of continuous homomorphisms ("characters"). They match the classical -sets if is discrete, and refine the homotopical compactness properties of Abels and Tiemeyer. Moreover, our theory recovers the definition of and proposed by Kochloukova. Besides presenting various characterizations of (particularly for ), we show that characters in are also in if is a closed cocompact subgroup, and we generalize several classical results. Namely, we prove that the set of nonzero elements of is open, we prove that characters in a group of type that do not vanish on the center always lie in , and we relate the -sets of a group with those of its quotients by closed subgroups of type . Lastly, we describe how governs whether a closed normal subgroup with abelian quotient is of type , generalizing one of the highlights of the classical theory.

Paper Structure

This paper contains 24 sections, 75 theorems, 167 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the article
  4. Preliminaries -- simplicial sets
  5. Free simplicial sets
  6. Semisimplicial approximation
  7. Homotopy groups
  8. Simplicial homotopies
  9. Filtrations and essential connectedness
  10. The homotopical $\Sigma$-sets
  11. Definition and basic properties
  12. Alternative filtrations
  13. Relation to the classical $\Sigma$-sets
  14. A recap of $\Sigma^n$
  15. Brown's criterion for characters
  16. ...and 9 more sections

Key Result

Theorem 1

If $G$ is a discrete group, then for every $n\in \mathbb{N}$, we have

Figures (5)

  • Figure 1: Converting a disk in $(G_\chi \cdot \operatorname{E} X^m)_\chi$ to a disk in $\Gamma(G,X \mid R_{3k})_\chi$. Triangles on the left give rise to disks whose boundary read relators in $R_{3k}$.
  • Figure 2: The cone $\mathcal{D}$ of $\mathcal{S}$. Edges of $\mathcal{S}$ get mapped to edges of $\Gamma(G,X)_\chi$ as dictated by $r$. Edges emanating from the cone point get mapped to edges with labels in $X^{\lfloor \tfrac{m}{2} \rfloor}$.
  • Figure 3: A collection of $k-1$ triangles in $\Gamma(G, C \mid R_3^C)_\chi$ exhibits the edge $g\cdot(1,x)$ as homotopic relative endpoints to the path at $g$ reading $w_x = x_1 \cdots x_k$.
  • Figure 4: Triangles in $\Gamma(G, C \mid R_3^C)_\chi$ exhibiting the loop $\gamma$ reading $r$ as null-homotopic. The orientation of edges labeled by $z_i$ should be understood to be reversed when $z_i\in \bar{C}$.
  • Figure 5: Bounding $\Vert T\Vert$. The shaded region indicates where $T$ might lie, once $P, h$ and $r$ have been fixed.

Theorems & Definitions (166)

  • Theorem 1: The classical $\Sigma^n$
  • Theorem 2: Brown's criterion for characters
  • Theorem 3: $\Sigma_\mathrm{top}^1$ and $\Sigma_\mathrm{top}^2$
  • Theorem 4: Openness
  • Theorem 5: Criteria for $\Sigma_\mathrm{top}^n$
  • Theorem 6: Closed cocompact subgroups
  • Theorem 7: Non-vanishing on the center
  • Theorem 8: $\Sigma_\mathrm{top}^n$ and quotients
  • Theorem 9: Co-abelian subgroups
  • Corollary 10: Compactness properties and quotients
  • ...and 156 more