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On the $K$-theory of groups of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ square-free

Luis Jorge Sánchez Saldaña, Mario Velásquez

Abstract

We provide an explicit computation of the topological $K$-theory groups $K_*(C_r^*(\mathbb{Z}^n\rtimes \mathbb{Z}/m))$ of semidirect products of the form $\mathbb{Z}^n\rtimes \mathbb{Z}Z/m$ with $m$ square-free. We want to highlight the fact that we are not impossing any conditions on the $\Z/m$-action on $\mathbb{Z}^n$. This generalizes previous computations of Lück-Davis and Langer-Lück.

On the $K$-theory of groups of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ square-free

Abstract

We provide an explicit computation of the topological -theory groups of semidirect products of the form with square-free. We want to highlight the fact that we are not impossing any conditions on the -action on . This generalizes previous computations of Lück-Davis and Langer-Lück.

Paper Structure

This paper contains 8 sections, 15 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classifying spaces for families and push-out constructions
  4. Splitting theorem of Angel-Becerra-Velásquez
  5. Torsion freeness of $K_*(C_r^*(\mathbb{Z}^n\rtimes G))$
  6. Torsion freeness of $K_*^{\mathbb{Z}^n\rtimes \mathbb{Z}/p}(\underline{E}(\mathbb{Z}^n\rtimes \mathbb{Z}/p))$
  7. Torsion freeness of $K_*^{\mathbb{Z}^n\rtimes G}(\underline{E}(\mathbb{Z}^n\rtimes G))$
  8. Formula for the rank

Key Result

Theorem 1.2

Let $h$ be a fixed generator of $G$, and let $t\geq 1$. Let $e_s$ be the $s$-th elementary symmetric polynomial in $n$-variables. Denote by $\bar{y}_{G}^t$ (resp. $\bar{y}_{G_p}^t$) the $n$-tuple of eigenvalues (including multiplicities) of $h^t\otimes \mathrm{Id}:\mathbb{Z}^n\otimes\mathbb C\to \ma where $k_l=\frac{p\left(p^{\frac{l}{p-1}}-1\right)}{m}$ and $l=n-\mathop{\mathrm{rank}}\nolimits((\

Theorems & Definitions (26)

  • Theorem 1.2: \ref{['main:rank:formula']}
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • ...and 16 more