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Register a new userSpectral theory of approximate lattices in nilpotent Lie groups
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We show that an approximate lattice in a nilpotent Lie group admits a relatively dense subset of central $(1-epsilon)$-Bragg peaks for every $epsilon > 0$. For the Heisenberg group we deduce that the union of horizontal and vertical $(1-epsilon)$-Bragg peaks is relatively dense in the unitary dual. More generally we study uniform approximate lattices in extensions of lcsc groups. We obtain necesary and sufficient conditions for the existence of a continuous horizontal factor of the associated hull-dynamical system, and study the spectral theory of the hull-dynamical system relative to this horizontal factor.
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Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $varphi$ be the local flow on $M$ induced by a $1$-parameter subgroup. Assume $K$ is a compact set of fixed points of $varphi$ and $U$ is a neighborhood of $K$ containing no other fixed points. Theorem: If the Dold fixed-point index of $varphi_t|U$ is nonzero for sufficiently small $t>0$, then ${rm Fix} (G) cap K e emptyset$.
We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural $mathrm{GL}(n,mathbb{R})$ action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature $s$ under gauge transformations of the metric, and show that Lie algebra derivations with nonzero trace obstruct the existence of Einstein metrics with $s eq0$. Using the notion of nice Lie algebra, we give the first example of a left-invariant Einstein metric with $s eq0$ on a nilpotent Lie group. We show that nilpotent Lie groups of dimension $leq 6$ do not admit such a metric, and a similar result holds in dimension $7$ with the extra assumption that the Lie algebra is nice.
We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension $n$ up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for $nleq9$. On every nilpotent Lie algebra of dimension $leq 7$, we determine the number of inequivalent nice bases, which can be $0$, $1$, or $2$. We show that any nilpotent Lie algebra of dimension $n$ has at most countably many inequivalent nice bases.
In this paper an alternative definition of the Rumin complex $(E_0^bullet,d_c)$ is presented, one that relies on a different concept of weights of forms. In this way, the Rumin complex can be constructed on any nilpotent Lie group equipped with a Carnot-Caratheodory metric. Moreover, this construction allows for the direct application of previous non-vanishing results of $ell^{q,p}$ cohomology to all nilpotent Lie groups that admit a positive grading.
We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the group C*-algebra of a finitely generated virtually nilpotent group $G$ is bounded by $2cdot h(G)!-1$, where $h(G)$ is the Hirsch length of $G.$ This extends and sharpens results of the first and third authors on finitely generated nilpotent groups. It then follows that if a C*-algebra generated by an irreducible representation of a virtually nilpotent group satisfies the universal coefficient theorem, it is classified by its Elliott invariant.
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