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 $d \colon X\times X \to {}^{n*}\R_{\ge0}$ satisfies the usual metric axioms (distance zero iff equality, symmetry, and triangle inequality).

Apartments are constructed from a real Euclidean space $V$ with a root system $\Phi$, extended to ${}^{n*}V = {}^{n*}\R\otimes_\R V$. Hyper-affine reflections

$$s_{a,k}(v) = v - 2\,\frac{{}^{n*}(a,v)-k}{(a,a)}\,a$$

for $a\in\Phi$, $k\in\Z^n$ generate the $n$-level Weyl group $W_n(\Phi) = W(\Phi)\ltimes \Z^n(\Phi^\vee)$. The fundamental chamber $C_0$ leads to the affine $\Z^n$-apartment

$$\Sigma(n, \Phi) = \bigcup_{w\in W_n(\Phi)} w\overline{C_0} \subset {}^{n*}V,$$

equipped with the hyper-Euclidean metric ${}^{n*}d_V$. Sectors and chambers are defined by fundamental domains for certain subgroups of $W_n(\Phi)$, stratified by level.

2. Babel Building Axioms and Construction

An $n$-level Babel building of type $\Phi$ is a pair $(X, \mathcal{A})$ where $\mathcal{A}$ is a collection of subsets of $X$, each isometric to $\Sigma(n, \Phi)$, such that:

1. For any two sectors $\mathcal{C}$ and $\mathcal{D}$, there exist subsectors $\mathcal{C}', \mathcal{D}'$ and some apartment $A \in \mathcal{A}$ with $\mathcal{C}'\cup\mathcal{D}'\subset A$.
2. Any two apartments admit a unique isometry fixing their intersection pointwise.

A canonical global metric $d_X\colon X\times X \to {}^{n*}\R_{\ge 0}$ exists, extending the metrics on apartments and ensuring triangularity. Retractions to apartments $\rho_{A,C}:X\to A$ are distance-decreasing.

The generalized CAT(0) inequality holds: for $p_t=(1-t)x + ty$ with $t\in{}^{n*}[0,1]$ and all $z\in X$,

$$d_X^2(p_t,z)\le (1-t)\,d_X^2(x,z) + t\,d_X^2(y,z) - t(1-t)\,d_X^2(x,y).$$

Residues encode the nested structure: for vertex $p\in X$,

$r_pX = \{x\in X\mid d_X(p,x)\in{}^{(n-1)*}\R \}$

with $r_pX$ itself an $(n-1)$-level Babel building. Iterating residues constructs a tower down to a classical affine building.

3. Metric and Connectivity Properties

Apartments $\Sigma(n, \Phi)$ are not convex in ${}^{n*}V$ for $n > 1$; for instance, the enclosure of two points $\cl(\{x,y\})$ can fragment into several disjoint affine sectors. The entire building $X$ is non-connected: vertices are equivalent ($p\sim q$) if $d_X(p,q)\in\R$. Thus,

$X = \bigsqcup_{[p]\in B(X)} r_p^{n-1}X$

decomposes $X$ into a disjoint union of affine buildings.

Although $(X, d_X)$ is not a CAT(0) space in the classical sense, it satisfies the ${}^{n*}\R$-valued CAT(0) inequality. If a group $G$ acts by isometries, stabilizing a bounded subset $B\subset X$ with a circumcenter, the circumcenter is unique and $G$-fixed.

4. Nesting Structure and Residues

The residue hierarchy provides a canonical chain: $$X = X_n \supset X_{n-1} = r_p X_n \supset \dots \supset X_1 = r_* X_2$$ with each $X_i$ an $i$-level Babel building. If $d_X(p,q)\in{}^{(n-1)*}\R$, then $r_p X = r_q X$; otherwise their intersection is empty. Sectors and apartments at level $i$ descend consistently to those at level $i-1$, with the property that $d(p, \cdot)\in{}^{(i-1)*}\R$.

5. Group Actions and Decompositions

Let $\widehat{G}$ act isometrically and strongly transitively on pairs (apartment $A$, chamber $C$). Define: $\widehat{B} = \Stab_{\widehat{G}}(C),\quad \widehat{N} = \Stab_{\widehat{G}}(A),\quad \widehat{H} = \widehat{B}\cap \widehat{N},\quad \widehat{W} = \widehat{N}/\widehat{H}.$ For any subset $\Omega\subset A$, fixers and pointwise stabilizers satisfy

$$\widehat{P}_\Omega = (\widehat{N}\cap \widehat{P}_\Omega)\cdot (\widehat{G}\cap \widehat{P}_\Omega),\quad \widehat{P}_\Omega^\dagger = (\widehat{N}\cap \widehat{P}_\Omega^\dagger)\cdot (\widehat{G}\cap \widehat{P}_\Omega^\dagger).$$

Double coset bijections obtain: $$Q\backslash Q\widehat{N} Q'/Q' \longrightarrow (\widehat{N}\cap Q)\backslash \widehat{W}/(\widehat{N}\cap Q').$$ Bruhat decomposition: $$\widehat{G} = \bigsqcup_{w\in \widehat{W}}\,\widehat{B} w \widehat{B}$$ and Cartan decomposition with $K = \widehat{P}_o$ and fundamental domain $\mathbf{D}$: $$\widehat{G} = \bigsqcup_{v\in \widehat{V}_{\mathbf{D}}} K v K$$ with $\widehat{V}_{\mathbf{D}}\subset \widehat{W}$.

Generalized Kapranov decompositions exist for each sector pair $(i,j)$: $$\widehat{G} = \bigsqcup_{w\in \widehat{W}} \widehat{B}^0_{\mathbf{D}_i}\,w\,\widehat{B}^0_{\mathbf{D}_j}$$ where $\widehat{B}^0_{\mathbf{D}_i} = \bigcup_{\text{$i$-sectors }C} \Stab_{\widehat{G}}(C)$.

If $x,y,z,u\in A$ are colinear with $y,z\in [x,u]$,

$$\widehat{P}_x\,\widehat{P}_u \cap \widehat{P}_y\,\widehat{P}_z = (\widehat{P}_x\cap\widehat{P}_y)\,(\widehat{P}_z\cap\widehat{P}_u).$$

For any vertex $p$, $$r_p\widehat{G} = \widehat{P}_{r_pX}^\dagger/\widehat{P}_{r_pX}$$ acts strongly transitively on $r_pX$ and inherits all higher-level decompositions.

6. Representative Examples

In rank 1, $\Sigma(2, A_1)\subset {}^{2*}\R$ is the union, via $W_2(A_1)$, of hyper-intervals $[0,1]$ under affine reflections: $$s:x\mapsto -x,\quad w_1:x\mapsto 2\omega_1-x,\quad w_2:x\mapsto 2\omega_2-x.$$ For type $A_2$ and $B_2$, apartments yield planar tilings from repeated $\R^2$-alcoves indexed by $\omega_2$-shifts.

For $G=SL_2(F)$ with $F = k((t_1))((t_2))$ (a 2-dimensional local field), the Weyl group $W_2(A_1)$ 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 ($n=1$)	Babel Building ($n>1$)
Metric space type	$\R$-valued, CAT(0), convex	${}^{n*}\R$-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).