Dynamics of strongly I-regular hyperbolic elements on affine buildings

TY - JOUR

T1 - Dynamics of strongly I-regular hyperbolic elements on affine buildings

AU - Ciobotaru, Corina-Gabriela

PY - 2024/7

Y1 - 2024/7

N2 - 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.

AB - 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.

UR - https://arxiv.org/abs/2407.10320

U2 - 10.48550/arXiv.2407.10320

DO - 10.48550/arXiv.2407.10320

M3 - Journal article

JO - Unpublished

JF - Unpublished

ER -