Patrick Eberlein (University of North Carolina)

Geometric and dynamical properties of 2-step nilpotent Lie groups

Abstract. Let $N$ be a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric, and let $\mathfrak{N}$ denote its Lie algebra. We say that $N$ has type $(p,q)$ if the commutator ideal of $\mathfrak{N}$ has dimension p and codimension $q$ . Let $\mathfrak{s}\mathfrak{o}$ (q, $\mathbb{R}$ ) denote the $q \times q$ skew symmetric matrices with real entries. If $N$ has type $(p,q)$ , then $\mathfrak{N}$ is isomorphic to $\mathfrak{N}' = \mathbb{R}^q \oplus W$ (direct sum), where $W$ is a p-dimensional subspace of $\mathfrak{s}\mathfrak{o}$ (q, $\mathbb{R}$ ) and $\mathfrak{N}$ ' has a simple bracket relation for which $[\mathbb{R}^q, \mathbb{R}^q]=W$ and $W$ lies in the center of $\mathfrak{N}$ . We begin by describing some left invariant first integrals for the geodesic flow on the unit tangent bundle SN. This is accomplished by using the canonical Poisson structure on $\mathfrak{N}$ and finding first integrals of the energy Hamiltonian vector field on $\mathfrak{N}$ . Linear and quadratic first integrals on $\mathfrak{N}$ are classified. One of the most interesting quadratic examples includes the Ricci tensor of N, regarded as a symmetric, bilinear form on $\mathfrak{N}$ . Let $\{C^1 ,\dots , C^p\}$ be the $q \times q$ skew symmetric structure matrices that arise from a basis of $\mathfrak{N}' = \mathbb{R}^q \oplus W$ that is a union of bases from $\mathbb{R}^q$ and $W$ . Let $V = \mathfrak{s}\mathfrak{o}(q,\mathbb{R}) \times\cdots \times \mathfrak{s}\mathfrak{o}(q,\mathbb{R})$ ($p$ times), and let $C$ be the element $(C^1, \dots , C^p)$ in $V$ . The group $GL(q,\mathbb{R}) \times GL(p,\mathbb{R})$ acts in a natural way on $V$ . We consider the existence of left invariant metrics $\langle , \rangle$ on $N$ with two types of special Ricci tensor, one of which is the property of being a first integral for the geodesic flow. Metrics $\langle , \rangle$ with these two types of Ricci tensor exist if and only if the $H$ orbits of $C$ in $V$ are closed, where $H = SL(q,\mathbb{R}) \times SL(p,\mathbb{R})$ and $SL(q,\mathbb{R})$ respectively. We describe a criterion due to R. Richardson and P. Slodowy for an H orbit in V to be closed, where H is a self adjoint subgroup of $GL(q,\mathbb{R}) \times GL(p,\mathbb{R})$ . $N$ admits a lattice $L$ if and only if $\mathfrak{N}$ admits a basis $\mathfrak{B}$ whose structure constants are integers ( $\mathbb{Z}$ -structure). The problem of finding Anosov diffeomorphisms on $L \backslash N$ reduces to the algebraic problem of finding automorphisms h of $\mathfrak{N}$ such that h leaves invariant $\mathbb{Z}$ -span( $\mathfrak{B}$ ) and $\det(h) = 1$ or $-1$ . We describe an existence result for Anosov diffeomorphisms due to J. Lauret, and we also describe some nonexistence results. In particular, the nonexistence results hold for generic examples of type (p,q), where $\mathfrak{N}' = \mathbb{R}^q \oplus W$ , unless the pair $(p,q)$ belongs to a small explicit list. Nonexistence also holds when $W$ is of Clifford type or when $W = \mathfrak{s}\mathfrak{o}(3,\mathbb{R})$ and $q$ is even. Rational structures on $\mathfrak{N}$ correspond bijectively to commensurability classes of lattices in $N$ . Two rational structures on $\mathfrak{N}$ are said to be equivalent if they differ by an automorphism of $\mathfrak{N}$ . If $\mathfrak{N}$ of type $(p,q)$ has a rational structure, then $\mathfrak{N}$ is isomorphic to $\mathbb{R}^q \oplus W$ , where $W$ has a basis of elements with rational entries. Moreover, $\mathbb{R}^q \oplus W$ admits a basis as discussed above where the entries of the structure element $C = (C^1, \dots , C^p)$ in $V = \mathfrak{s}\mathfrak{o}(q,\mathbb{R}) \times\cdots \times \mathfrak{s}\mathfrak{o}(q,\mathbb{R})$ ($p$ times) are rational. Let $G$ denote $GL(p,\mathbb{R}) \times GL(q,\mathbb{R})$ and let $G_{Q}$ denote $GL(p,\mathbb{Q}) \times GL(q,\mathbb{Q})$ . The space of rational structures on $\mathfrak{N}$ can be identified with $V(Q,C) / G_{Q}$ , where $V(Q,C)$ consists of the elements with rational entries in the orbit $G(C)$ in $V$ .