Dynamics of strongly I-regular hyperbolic elements on affine buildings

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 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 ∂_∞ Δ of Δ is much more complicated than for the strongly regular ones, the limn→ ∞ γⁿ(ξ) still exists in ∂_∞ Δ for ideal points ξ ∈ ∂_∞ Δ that satisfy certain assumptions. An important role in this business is played by the cone topology on Δ ∪ ∂_∞ Δ and the projection of specific residues of ∂_∞ 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.