Babel Buildings: Higher-Dimensional Structures

• Babel buildings are higher-dimensional generalizations of affine buildings characterized by non-connected, non-convex metric spaces and intricate residue hierarchies.
• They leverage hyper-real fields, nested apartments, and higher-level Weyl group actions to analyze groups over multidimensional local fields.
• The framework facilitates novel group decompositions, generalized CAT(0) inequalities, and iterative residues that descend to classical affine buildings.

Babel buildings are a higher-dimensional generalization of affine buildings, forming non-connected, non-convex metric spaces equipped with a framework that supports intricate group actions and decompositions. The construction leverages hyper-real fields of arbitrary finite level, nested apartments, and a distinctive residue hierarchy, resulting in a rich geometric and combinatorial structure suitable for the analysis of groups over higher-dimensional local fields (Mori, 21 Dec 2025).

1. Hyper-real Fields, Metric Spaces, and Apartments

The theory is based on the construction of hyper-real fields of level $n$, ${}^{n*}\R$, using repeated ultraproducts via a nonprincipal ultrafilter $\mathcal{F}\supset\{\text{cofinite sets}\}$: $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$ Iterating this construction, one obtains ${}^{n*}\R$ as a totally ordered field which contains a lex-ordered lattice $$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$, with each $\omega_i$ infinitely large over ${}^{(i-1)*}\R$. An ${}^{n*}\R$-metric space $(X, d)$ requires that ${}^{n*}\R$0 satisfies the usual metric axioms (distance zero iff equality, symmetry, and triangle inequality).

Apartments are constructed from a real Euclidean space ${}^{n*}\R$1 with a root system ${}^{n*}\R$2, extended to ${}^{n*}\R$3. Hyper-affine reflections

${}^{n*}\R$4

for ${}^{n*}\R$5, ${}^{n*}\R$6 generate the ${}^{n*}\R$7-level Weyl group ${}^{n*}\R$8. The fundamental chamber ${}^{n*}\R$9 leads to the affine $\mathcal{F}\supset\{\text{cofinite sets}\}$0-apartment

$\mathcal{F}\supset\{\text{cofinite sets}\}$1

equipped with the hyper-Euclidean metric $\mathcal{F}\supset\{\text{cofinite sets}\}$2. Sectors and chambers are defined by fundamental domains for certain subgroups of $\mathcal{F}\supset\{\text{cofinite sets}\}$3, stratified by level.

2. Babel Building Axioms and Construction

An $\mathcal{F}\supset\{\text{cofinite sets}\}$4-level Babel building of type $\mathcal{F}\supset\{\text{cofinite sets}\}$5 is a pair $\mathcal{F}\supset\{\text{cofinite sets}\}$6 where $\mathcal{F}\supset\{\text{cofinite sets}\}$7 is a collection of subsets of $\mathcal{F}\supset\{\text{cofinite sets}\}$8, each isometric to $\mathcal{F}\supset\{\text{cofinite sets}\}$9, such that:

1. For any two sectors $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$0 and $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$1, there exist subsectors $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$2 and some apartment $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$3 with $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$4.
2. Any two apartments admit a unique isometry fixing their intersection pointwise.

A canonical global metric $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$5 exists, extending the metrics on apartments and ensuring triangularity. Retractions to apartments $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$6 are distance-decreasing.

The generalized CAT(0) inequality holds: for $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$7 with $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$8 and all $${}^*A = A^{\mathbb N}/\sim_\mathcal{F},\quad [a_m] \sim [b_m] \iff \{m\mid a_m = b_m\} \in \mathcal{F}.$$9,

${}^{n*}\R$0

Residues encode the nested structure: for vertex ${}^{n*}\R$1,

${}^{n*}\R$2

with ${}^{n*}\R$3 itself an ${}^{n*}\R$4-level Babel building. Iterating residues constructs a tower down to a classical affine building.

3. Metric and Connectivity Properties

Apartments ${}^{n*}\R$5 are not convex in ${}^{n*}\R$6 for ${}^{n*}\R$7; for instance, the enclosure of two points ${}^{n*}\R$8 can fragment into several disjoint affine sectors. The entire building ${}^{n*}\R$9 is non-connected: vertices are equivalent ($$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$0) if $$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$1. Thus,

$$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$2

decomposes $$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$3 into a disjoint union of affine buildings.

Although $$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$4 is not a CAT(0) space in the classical sense, it satisfies the $$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$5-valued CAT(0) inequality. If a group $$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$6 acts by isometries, stabilizing a bounded subset $$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$7 with a circumcenter, the circumcenter is unique and $$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$8-fixed.

4. Nesting Structure and Residues

The residue hierarchy provides a canonical chain: $$\Z^n \simeq \bigoplus_{i=1}^n \Z\,\omega_i \subset {}^{n*}\R$$9 with each $\omega_i$0 an $\omega_i$1-level Babel building. If $\omega_i$2, then $\omega_i$3; otherwise their intersection is empty. Sectors and apartments at level $\omega_i$4 descend consistently to those at level $\omega_i$5, with the property that $\omega_i$6.

5. Group Actions and Decompositions

Let $\omega_i$7 act isometrically and strongly transitively on pairs (apartment $\omega_i$8, chamber $\omega_i$9). Define: ${}^{(i-1)*}\R$0 For any subset ${}^{(i-1)*}\R$1, fixers and pointwise stabilizers satisfy

${}^{(i-1)*}\R$2

Double coset bijections obtain: ${}^{(i-1)*}\R$3 Bruhat decomposition: ${}^{(i-1)*}\R$4 and Cartan decomposition with ${}^{(i-1)*}\R$5 and fundamental domain ${}^{(i-1)*}\R$6: ${}^{(i-1)*}\R$7 with ${}^{(i-1)*}\R$8.

Generalized Kapranov decompositions exist for each sector pair ${}^{(i-1)*}\R$9: ${}^{n*}\R$0 where ${}^{n*}\R$1i${}^{n*}\R$2.

If ${}^{n*}\R$3 are colinear with ${}^{n*}\R$4,

${}^{n*}\R$5

For any vertex ${}^{n*}\R$6, ${}^{n*}\R$7 acts strongly transitively on ${}^{n*}\R$8 and inherits all higher-level decompositions.

6. Representative Examples

In rank 1, ${}^{n*}\R$9 is the union, via $(X, d)$0, of hyper-intervals $(X, d)$1 under affine reflections: $(X, d)$2 For type $(X, d)$3 and $(X, d)$4, apartments yield planar tilings from repeated $(X, d)$5-alcoves indexed by $(X, d)$6-shifts.

For $(X, d)$7 with $(X, d)$8 (a 2-dimensional local field), the Weyl group $(X, d)$9 realizes the group-theoretic structure: \begin{align*} G &= \bigsqcup_{w\in W_2(A_1)} B w B,\ G &= \bigsqcup_{v\in\Z^2_{\ge0}} K\,\diag(t_1^{v_1}t_2^{v_2},\,t_1^{-v_1}t_2^{-v_2})\,K, \end{align*} and relevant Kapranov decompositions.

Table: Structural Features of Babel Buildings

Feature	Affine Building (${}^{n*}\R$00)	Babel Building (${}^{n*}\R$01)
Metric space type	${}^{n*}\R$02-valued, CAT(0), convex	${}^{n*}\R$03-valued, non-convex, non-connected
Apartments	Affine spaces	Lex-ordered hyper-apartments
Decomposition towers	No further nesting	Nested chain down to affine building
Group decompositions	Bruhat, Cartan (classical)	Higher-level Bruhat, Cartan, Kapranov

The Babel building framework provides a canonical geometric setting for analyzing group actions and decompositions associated with groups over multidimensional local fields, generalizing and extending the role of classical buildings to non-connected, stratified, hyper-metric spaces (Mori, 21 Dec 2025).