Dynamics of actions of automorphisms of discrete groups G G G on S u b G \mathrm{Sub}_G Sub G ​ and applications to lattices in Lie groups

[摘要] For a locally compact Hausdorff group GGG and the compact space SubG\mathrm{Sub}_GSubG​ of closed subgroups of GGG endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of GGG on SubG\mathrm{Sub}_GSubG​ in terms of distality and expansivity. We prove that an infinite discrete group GGG, which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on SubGc\mathrm{Sub}^c_GSubGc​, the space of cyclic subgroups of GGG, while only the finite order automorphisms of GGG act distally on SubGc\mathrm{Sub}^c_GSubGc​. For an automorphism TTT of a connected Lie group GGG which keeps a lattice Γ\GammaΓ invariant, we compare the behaviour of the actions of TTT on SubG\mathrm{Sub}_GSubG​ and SubΓ\mathrm{Sub}_\GammaSubΓ​ in terms of distality. Under certain necessary conditions on the Lie group GGG, we show that TTT acts distally on SubG\mathrm{Sub}_GSubG​ if and only if it acts distally on SubΓ\mathrm{Sub}_\GammaSubΓ​. We also obtain certain results about the structure of lattices in a connected Lie group.

[学科分类] 神经科学

[关键词] Distal and expansive actions of automorphisms;space of closed subgroups;Chabauty topology;polycyclic groups;lattices in connected Lie groups