Dynamics of strongly I-regular hyperbolic elements on affine buildings

TL;DR

This paper studies a new class of hyperbolic automorphisms called strongly I-regular on affine buildings, exploring their dynamics and existence, and applies these results to understand limits of subgroups in groups acting on these buildings.

Contribution

It introduces strongly I-regular hyperbolic automorphisms for affine buildings, proves their existence, analyzes their boundary dynamics, and relates these to subgroup limits in automorphism groups.

Findings

Strongly I-regular hyperbolic automorphisms exist in cocompact group actions.

The boundary limit behavior of these automorphisms is characterized under certain conditions.

Chabauty limits of subgroups contain unipotent radicals of parabolic subgroups.

Abstract

The first goal of this article is to investigate a refinement of previously-introduced strongly regular hyperbolic automorphisms of locally finite thick Euclidean buildings $\Delta$ of finite Coxeter system $(W,S)$. The new ones are defined for each proper subset $I \subsetneq S$ and called strongly $I$-regular hyperbolic automorphisms of $\Delta$. Generalizing previous results, we show that such elements exist in any group $G$ acting cocompactly and by automorphisms on $\Delta$. Although the dynamics of strongly $I$-regular hyperbolic elements $\gamma$ on the spherical building $\partial_\infty \Delta$ of $\Delta$ is much more complicated than for the strongly regular ones, the $\lim\limits_{n\to \infty} \gamma^{n}(\xi)$ still exists in $\partial_\infty \Delta$ for ideal points $\xi \in \partial_\infty \Delta$ that satisfy certain assumptions. An important role in this business is played by the cone topology on $\Delta \cup \partial_\infty \Delta$ and the projection of specific residues of $\partial_\infty \Delta$ on the ideal boundary of $Min(\gamma)$. All the above research is performed in order to achieve the second, and main, goal of the article. Namely, we prove that for closed groups $G$ with a type-preserving and strongly transitive action by automorphisms on $\Delta$, the Chabauty limits of certain closed subgroups of $G$ contain as a normal subgroup the entire unipotent radical of concrete parabolic subgroups of $G$.