[Paper Review] Polygons in buildings and their refined side lengths

Michael Kapovich, Bernhard Leeb+1 more|ArXiv.org|Jun 15, 2004

Geometric and Algebraic Topology7 references4 citations

TL;DR

This paper establishes that the set of $Δ_{euc}$-valued side lengths of oriented $n$-gons in thick Euclidean buildings depends only on the spherical Weyl chamber $Δ_{sph}$ of the building’s Tits boundary. Using a Gauss map construction, it proves that such side lengths correspond precisely to the $Δ_{euc}$-weights of semistable weighted configurations on the Tits boundary, linking geometric polygon constraints to stability conditions in geometric invariant theory.

ABSTRACT

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber; in addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the vector valued side lengths of polygons in (thick) Euclidean buildings. The main result is that the set of possible side lengths of n-gons depends only on the associated spherical Coxeter complex. We show moreover that it coincides with the space of vector valued weights of semistable weighted configurations on the Tits boundary. The side lengths of polygons in symmetric spaces of noncompact type are studied in the related paper [KLM1]. Applications of the geometric results in both papers to algebraic group theory are given in [KLM3].

Motivation & Objective

• To characterize the set of $Δ_{euc}$-valued side lengths of oriented $n$-gons in thick Euclidean buildings.
• To establish a correspondence between such polygons and weighted configurations on the Tits boundary at infinity.
• To show that the existence of a polygon with given side lengths is equivalent to the existence of a semistable weighted configuration with matching $Δ_{euc}$-weights.
• To prove that the space of possible side lengths depends only on the spherical Coxeter complex, not on the specific building.
• To extend results from symmetric spaces of noncompact type to thick Euclidean buildings via a unified stability-theoretic framework.

Proposed method

• Define the $Δ$-length of a geodesic segment as a vector in the Euclidean Weyl chamber $Δ_{euc}$, encoding both metric length and direction.
• Introduce a Gauss map construction that associates an oriented $n$-gon in a Euclidean building to a weighted configuration on its Tits boundary $Δ_{sph}$.
• Define semistability for weighted configurations on spherical buildings using asymptotic properties of weighted Busemann functions $b_{ψ}$.
• Use the transfer theorem to lift polygon existence from the cone over the Tits boundary to the original building.
• Apply fixed-point theory for weak contractions to show that semistable configurations arise as Gauss maps of actual polygons.
• Use discrete approximations of the Tits boundary and bounded sublevel sets of Busemann functions to control convergence and existence.

Experimental results

Research questions

• RQ1 What constraints govern the $Δ_{euc}$-valued side lengths of oriented $n$-gons in thick Euclidean buildings?
• RQ2 How are these side lengths related to configurations on the Tits boundary at infinity?
• RQ3 Can the existence of a polygon with given $Δ$-side lengths be characterized via stability conditions on the boundary?
• RQ4 Does the space of possible $Δ$-side lengths depend only on the spherical Weyl chamber, or on the full building structure?
• RQ5 What is the precise relationship between semistable weighted configurations on the Tits boundary and actual polygons in the building?

Key findings

• The set ${\cal P}_n(X)$ of $Δ_{euc}$-valued side lengths of oriented $n$-gons in a thick Euclidean building $X$ depends only on the spherical Weyl chamber $Δ_{sph}$ of its Tits boundary.
• A polygon with $Δ$-side lengths $h$ exists in $X$ if and only if there exists a semistable weighted configuration on $Δ_{sph}$ with $Δ_{euc}$-weights $h$.
• The space ${\cal P}_n(\Delta_{sph})$ of such side lengths is a finite-sided convex polyhedral cone for spherical Coxeter complexes arising from symmetric spaces.
• For spherical Coxeter complexes not arising from symmetric spaces (e.g., dihedral group $D_8$), the structure of ${\cal P}_n(\Delta_{sph})$ remains open, though the stability framework still applies.
• The existence of fixed points for weak contractions on sublevel sets of Busemann functions confirms that semistable configurations arise as Gauss maps of polygons.
• The transfer theorem allows lifting polygon existence from the cone over the Tits boundary to the original building, preserving $Δ$-side lengths.