Dynamics of strongly I-regular hyperbolic elements on affine buildings

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 $Δ$ 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 $Δ$. Generalizing previous results, we show that such elements exist in any group $G$ acting cocompactly and by automorphisms on $Δ$. Although the dynamics of strongly $I$-regular hyperbolic elements $γ$ on the spherical building $\partial_\infty Δ$ of $Δ$ is much more complicated than for the strongly regular ones, the $\lim\limits_{n\to \infty} γ^{n}(ξ)$ still exists in $\partial_\infty Δ$ for ideal points $ξ\in \partial_\infty Δ$ that satisfy certain assumptions. An important role in this business is played by the cone topology on $Δ\cup \partial_\infty Δ$ and the projection of specific residues of $\partial_\infty Δ$ on the ideal boundary of $Min(γ)$. 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 $Δ$, the Chabauty limits of certain closed subgroups of $G$ contain as a normal subgroup the entire unipotent radical of concrete parabolic subgroups of $G$.