Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups

Abstract

We study the existence of various types of gradings on Lie algebras, such as Carnot gradings or gradings in positive integers, and prove that the existence of such gradings is invariant under extensions of scalars.As an application, we prove that if Γ is a finitely generated nilpotent group, its systolic growth is asymptotically equivalent to its word growth if and only if the Malcev completion of Γ is Carnot.We also characterize when Γ is non-cohopfian, in terms of the existence of a non-trivial grading in non-negative integers, and deduce that this property only depends on its real (or even complex) Malcev completion. n≥0 φ n (Γ) = {1}; such φ is called a dis-cohopf endomorphism. It appears, for a nontrivial group, as a strong negation of being cohopfian. Contents

Formality properties of finitely generated groups and Lie algebras

“…As we shall see in Proposition 9.3, the above theorem generalizes a recent result of Cornulier (Theorem 3.14 from [13]).…”

“…A classical example is the unipotent group U n (Z), which is known to be filtered-formal by Lambe and Priddy [42], but not graded-formal for n ≥ 3. In [13], Cornulier showed that the filtered-formality of a finite-dimensional nilpotent Lie algebra is independent of the ground field, thereby answering a question of Johnson [37]. The much more general Theorem 1.1 allows us to recover this result in Proposition 9.3.…”

“…Carlson and Toledo [8] studied the 1-formality properties of such groups, while Plantiko [65] gave a sufficient conditions for such groups to be non-graded-formal. For nilpotent Lie algebras, the notion of filtered-formality has been studied by Leger [46], Cornulier [13], Kasuya [38], and others. In particular, it is shown in [13] that the systolic growth of a finitely generated nilpotent group G is asymptotically equivalent to its growth if and only if the Malcev Lie algebra m(G; k) is filtered-formal (or, 'Carnot'), while in [38] it is shown that the variety of flat connections on a filtered-formal (or, 'naturally graded'), n-step nilpotent Lie algebra has a singularity at the origin cut out by polynomials of degree at most n + 1.…”

Pro-groups and generalizations of a theorem of Bing

“…For example, Belegradek considered in [8] when such a lattice must be co-Hopfian, and in particular when they are not. Non co-Hopfian subgroups of nilpotent Lie groups were also studied by Dekimpe, Lee and Potyagailo in [22,23,24], and by Cornulier in [20]. Here is a general version of Problem 8.1:…”

Gradings on Lie algebras with applications to infra-nilmanifolds

“…As a consequence of this theorem we construct a nilmanifold admitting an Anosov diffeomorphism but no expanding map in the last section. Some of these results were proved independently and by different methods in [2] by Y. Cornulier. More detailed references to the work of Y. Cornulier are given throughout paper.…”