Algebraic Anosov actions of nilpotent Lie groups (Q1928431)

The algebraic Anosov actions considered in this paper are given by \((G,K,\Gamma,{\mathcal H})\) where \(G\) is a connected Lie group, \(K\) a compact subgroup of \(G\), \(\Gamma\) a torsion-free uniform lattice in \(G\), and \({\mathcal H}\) a subalgebra of the Lie algebra \({\mathcal G}\) of \(G\) contained in the normalizer of the Lie algebra \({\mathcal K}\) tangent to \(K\). For the simply connected Lie group \(H\) with Lie algebra isomorphic to \({\mathcal H}\), an action of \(H\) is induced on \(\Gamma\setminus G/K\) by \({\mathcal H}\). The authors in this paper classify algebraic Anosov actions \((G,K,\Gamma,{\mathcal H})\) when \(H\) is nilpotent. This classification is an extension of \textit{P. Tomter}'s classification of algebraic Anosov flows [Global Analysis, Proc. Sympos. Pure Math. 14, 299--327 (1970; Zbl 0207.54502); Topology 14, 179--189 (1975; Zbl 0365.58013)] to the nilpotent case. An Anosov action where \(G\) is a solvable Lie group, is commensurate to a nil-suspension over the suspension of an Anosov \({\mathbb Z}^p\) action on the torus. An algebraic Anosov action where \(G\) is a semisimple Lie group, is commensurate to a modified Weyl chamber action. An algebraic Anosov action where \(G\) is non-solvable and non-semisimple, is commensurate to one of three things: (1) an algebraic Anosov action on a solvable Lie group, (2) a central extension over a (modified) Weyl chamber action, or (3) a nil-suspension over an algebraic Anosov action which is commensurate to a reductive algebraic Anosov action.