Fixed points for group actions on 2-dimensional affine buildings

TL;DR

This paper proves a local-to-global fixed point theorem for groups acting on 2-dimensional affine buildings of types A2 and C2, confirming special cases of Marquis's conjecture in the discrete setting.

Contribution

It establishes a new local-to-global fixed point criterion for group actions on affine buildings of specific types, advancing understanding of group actions in geometric group theory.

Findings

Proves fixed point properties for groups on affine buildings of types A2 and C2.

Confirms special cases of Marquis's conjecture in the discrete case.

Provides a unified approach for non-discrete and discrete affine buildings.

Abstract

We prove a local-to-global result for fixed points of groups acting on affine buildings (possibly non-discrete) of types $\tilde{A}_2$ or $\tilde{C}_2$. In the discrete case, our theorem establishes the corresponding special cases of a conjecture by Marquis.