Geodesic
Dynamics of actions of automorphisms of discrete groups $G$ on $\mathrm{Sub}_G$ and applications to lattices in Lie groups
Rajdip Palit1 ; Manoj B. Prajapati2 ; Riddhi Shah1
1Jawaharlal Nehru University, New Delhi, India
2Jawaharlal Nehru University, New Delhi; Charotar University of Science and Technology (CHARUSAT), India
Groups, geometry, and dynamics, Tome 17 (2023) no. 1, pp. 185-213

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For a locally compact Hausdorff group G and the compact space SubG​ of closed subgroups of G endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of G on SubG​ in terms of distality and expansivity. We prove that an infinite discrete group G, which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on SubGc​, the space of cyclic subgroups of G, while only the finite order automorphisms of G act distally on SubGc​. For an automorphism T of a connected Lie group G which keeps a lattice Γ invariant, we compare the behaviour of the actions of T on SubG​ and SubΓ​ in terms of distality. Under certain necessary conditions on the Lie group G, we show that T acts distally on SubG​ if and only if it acts distally on SubΓ​. We also obtain certain results about the structure of lattices in a connected Lie group.
DOI : 10.4171/ggd/672
Classification : 37-XX, 22-XX
Mots-clés : Distal and expansive actions of automorphisms, space of closed subgroups, Chabauty topology, polycyclic groups, lattices in connected Lie groups

Rajdip Palit  1   ; Manoj B. Prajapati  2   ; Riddhi Shah  1

1 Jawaharlal Nehru University, New Delhi, India
2 Jawaharlal Nehru University, New Delhi; Charotar University of Science and Technology (CHARUSAT), India 
Rajdip Palit; Manoj B. Prajapati; Riddhi Shah. Dynamics of actions of automorphisms of discrete groups $G$ on $\mathrm{Sub}_G$ and applications to lattices in Lie groups. Groups, geometry, and dynamics, Tome 17 (2023) no. 1, pp. 185-213. doi: 10.4171/ggd/672
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     title = {Dynamics of actions of automorphisms of discrete groups $G$ on $\mathrm{Sub}_G$ and applications to lattices in {Lie} groups},
     journal = {Groups, geometry, and dynamics},
     pages = {185--213},
     year = {2023},
     volume = {17},
     number = {1},
     doi = {10.4171/ggd/672},
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