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Nested Affine Buildings and Their Group Decompositions

Masaoki Mori

TL;DR

The paper extends Bruhat-Tits building theory to higher-dimensional local fields by introducing Babel buildings, spaces equipped with a $^{n*}R$-valued metric built from nested lower-level buildings. It first constructs a coherent metric framework with distance-decreasing retractions and a hyper-real CAT(0)-type inequality, enabling fixed-point-type results and a Kapranov-style decomposition in this setting. It then develops a group-action theory: strongly transitive actions yield Bruhat, Cartan, and Kapranov decompositions adapted to Babel buildings, with a detailed analysis of fixers, stabilizers, and double cosets. Residues at vertices provide a hierarchical reduction to $(n-1)$-level Babel buildings, linking global geometry to local (residue) structure and enabling potential homological study of arithmetic quotients in this higher-rank context.

Abstract

In this paper, we construct a higher dimensional generalization of affine buildings and introduce a new structure, which we call Babel buildings. These buildings are non-connected, non-convex metric spaces of non-positive curvature. Despite their non-standard properties, Babel buildings provide an effective framework for studying the structure of groups acting on them. We analyze the metric and nesting structures of Babel buildings and derive key results regarding the group actions consistent with this new framework.

Nested Affine Buildings and Their Group Decompositions

TL;DR

The paper extends Bruhat-Tits building theory to higher-dimensional local fields by introducing Babel buildings, spaces equipped with a -valued metric built from nested lower-level buildings. It first constructs a coherent metric framework with distance-decreasing retractions and a hyper-real CAT(0)-type inequality, enabling fixed-point-type results and a Kapranov-style decomposition in this setting. It then develops a group-action theory: strongly transitive actions yield Bruhat, Cartan, and Kapranov decompositions adapted to Babel buildings, with a detailed analysis of fixers, stabilizers, and double cosets. Residues at vertices provide a hierarchical reduction to -level Babel buildings, linking global geometry to local (residue) structure and enabling potential homological study of arithmetic quotients in this higher-rank context.

Abstract

In this paper, we construct a higher dimensional generalization of affine buildings and introduce a new structure, which we call Babel buildings. These buildings are non-connected, non-convex metric spaces of non-positive curvature. Despite their non-standard properties, Babel buildings provide an effective framework for studying the structure of groups acting on them. We analyze the metric and nesting structures of Babel buildings and derive key results regarding the group actions consistent with this new framework.

Paper Structure

This paper contains 8 sections, 27 theorems, 84 equations, 1 figure.

Table of Contents

  1. The structure of Babel buildings
  2. Hyper-extensions of sets
  3. Babel apartments
  4. The metric structure of Babel buildings and residues at vertices
  5. Group actions on Babel buildings
  6. Fixers and stabilizers
  7. Double cosets
  8. Residues at vertices

Key Result

Theorem 1

Let $X$ be an $n$-level Babel building. There exists a canonical distance function $d_X \colon X \times X \longrightarrow {}^{n*}\mathbb{R}$ which satisfies the triangle inequality. Furthermore, the retraction map onto any Babel apartment is distance-decreasing.

Figures (1)

  • Figure 4: ${\rm cl}(\{x,y\})$ in $\Sigma(2,A_{2})$ is inside the parallelogram above.

Theorems & Definitions (82)

  • Theorem 1: Main Theorem A
  • Theorem 2: Main Theorem B
  • Theorem 3: Main Theorem C
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Remark 1.6
  • ...and 72 more