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). For each proper subset \(I \subsetneq S\) , we define a new class of automorphisms 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 at infinity \(\partial _\infty \Delta\) are more intricate than of strongly regular ones, the \(\lim \limits _{n\rightarrow \infty } \gamma ^{n}(\xi )\) still exists in \(\partial _\infty \Delta\) for ideal points \(\xi \in \partial _\infty \Delta\) satisfying suitable conditions. A central role in this analysis is played by the cone topology on \(\Delta \cup \partial _\infty \Delta\) and the projection of specific residues in \(\partial _\infty \Delta\) onto the ideal boundary of \(\textrm{Min}(\gamma )\) . These investigations serve the second – and primary – goal of the article. Specifically, we prove that for closed groups G acting type-preservingly and strongly transitively on \(\Delta\) , the Chabauty limits of certain closed subgroups of G contain as a normal subgroup the full unipotent radical of explicit parabolic subgroups of G.
