Subcomplexes and fixed point sets of isometries of spherical buildings

TL;DR

This paper investigates the structure of convex subcomplexes and fixed point sets of isometries in spherical buildings, proving a conjecture for certain types and revealing geometric properties related to their circumradius.

Contribution

It establishes that top-dimensional fixed point sets are either subbuildings or have bounded circumradius, confirming a conjecture for types A_n and D_n spherical buildings.

Findings

Fixed point sets of top-dimensional isometries are either subbuildings or have circumradius ≤ π/2.

The results confirm the Kleiner-Leeb conjecture for types A_n and D_n.

Convex subcomplexes of spherical buildings exhibit specific geometric constraints.

Abstract

In this paper we study convex subcomplexes of spherical buildings. We pay special attention to fixed point sets of type-preserving isometries of spherical buildings. This sets are also convex subcomplexes of the natural polyhedral structure of the building. We show, among other things, that if the fixed point set is top-dimensional then it is either a subbuilding or it has circumradius $\leq \frac{\pi}{2}$. If the building is of type $A_n$ or $D_n$, we also show that the same conclusion holds for an arbitrary (top-dimensional in the $D_n$-case) convex subcomplex. This proves a conjecture of Kleiner-Leeb in these cases.