scholarly journals Nilpotent Lie Algebras and Systolic Growth of Nilmanifolds

2017 ◽  
Vol 2019 (9) ◽  
pp. 2763-2799
Author(s):  
Yves Cornulier
Keyword(s):  
Lie Algebras ◽  
Lie Group ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Growth Function ◽  
Nilpotent Groups ◽  
Upper Bounds ◽  
Nilpotent Lie Algebras ◽  
Open Questions ◽  
Simply Connected

Abstract Introduced by Gromov in the nineties, the systolic growth of a Lie group gives the smallest possible covolume of a lattice with a given systole. In a simply connected nilpotent Lie group, this function has polynomial growth, but can grow faster than the volume growth. We express this systolic growth function in terms of discrete cocompact subrings of the Lie algebra, making it more practical to estimate. After providing some general upper bounds, we develop methods to provide nontrivial lower bounds. We provide the first computations of the asymptotics of the systolic growth of nilpotent groups for which this is not equivalent to the volume growth. In particular, we provide an example for which the degree of growth is not an integer; it has dimension 7. Finally, we gather some open questions.

scholarly journals DISTAL ACTIONS OF AUTOMORPHISMS OF NILPOTENT GROUPS G ON SUB G AND APPLICATIONS TO LATTICES IN LIE GROUPS

2020 ◽  
pp. 1-20
Author(s):  
RAJDIP PALIT ◽  
RIDDHI SHAH
Keyword(s):  
Lie Group ◽  
Compact Subgroup ◽  
Nilpotent Groups ◽  
Discrete Groups ◽  
Nilpotent Lie Group ◽  
The Difference ◽  
Simply Connected ◽  
Chabauty Topology ◽  
Solvable Lie Group ◽  
Compactly Generated

Abstract For a locally compact group G, we study the distality of the action of automorphisms T of G on Sub G , the compact space of closed subgroups of G endowed with the Chabauty topology. For a certain class of discrete groups G, we show that T acts distally on Sub G if and only if T n is the identity map for some $n\in\mathbb N$ . As an application, we get that for a T-invariant lattice Γ in a simply connected nilpotent Lie group G, T acts distally on Sub G if and only if it acts distally on SubΓ. This also holds for any closed T-invariant co-compact subgroup Γ in G. For a lattice Γ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on SubΓ. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group and that in a nilpotent Lie group. We also characterise automorphisms of a lattice Γ in a connected semisimple Lie group which act distally on SubΓ. For torsion-free compactly generated nilpotent (metrisable) groups G, we obtain the following characterisation: T acts distally on Sub G if and only if T is contained in a compact subgroup of Aut(G). Using these results, we characterise the class of such groups G which act distally on Sub G . We also show that any compactly generated distal group G is Lie projective.

LEFT INVARIANT COMPLEX STRUCTURES ON NILPOTENT SIMPLY CONNECTED INDECOMPOSABLE 6-DIMENSIONAL REAL LIE GROUPS

2007 ◽  
Vol 17 (01) ◽  
pp. 115-139 ◽  
Author(s):  
L. MAGNIN
Keyword(s):  
Automorphism Group ◽  
Normal Form ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Normal Forms ◽  
Equivalence Classes ◽  
Complex Structures ◽  
Simply Connected ◽  
Connected Lie Group

Integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras have been computed in a previous paper, along with normal forms for representatives of the various equivalence classes under the action of the automorphism group. Here we go to the connected simply connected Lie group G0 associated to such a Lie algebra 𝔤. For each normal form J of integrable complex structures on 𝔤, we consider the left invariant complex manifold G = (G0, J) associated to G0 and J. We explicitly compute a global holomorphic chart for G and we write down the multiplication in that chart.

Lattices in a split solvable Lie group

1997 ◽  
Vol 122 (2) ◽  
pp. 245-250 ◽  
Author(s):  
RICHARD MOSAK ◽  
MARTIN MOSKOWITZ
Keyword(s):  
Euclidean Space ◽  
Lie Group ◽  
Nilpotent Groups ◽  
Integer Lattice ◽  
Heisenberg Groups ◽  
Simply Connected ◽  
Positive Integers ◽  
Solvable Lie Group

Given a Lie group, it is often useful to have a parametrization of the set of its lattices. In Euclidean space ℝn, for example, each lattice corresponds to a basis, and any lattice is equivalent to the standard integer lattice under an automorphism in GL(n, ℝ). In the nilpotent case, the lattices of the Heisenberg groups are classified, up to automorphisms, by certain sequences of positive integers with divisibility conditions (see [1]). In this paper we will study the set of lattices in a class of simply connected, solvable, but not nilpotent groups G. The construction of G depends on a diagonal n×n matrix Δ with distinct non-zero eigenvalues, of trace 0; we will writeformula here

scholarly journals Graphs and metric 2-step nilpotent Lie algebras

2018 ◽  
Vol 18 (3) ◽  
pp. 265-284 ◽  
Author(s):  
Rachelle C. DeCoste ◽  
Lisa DeMeyer ◽  
Meera G. Mainkar
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Inner Product ◽  
Nilpotent Lie Group ◽  
Closed Geodesics ◽  
Nilpotent Lie Algebra ◽  
Comprehensive Description ◽  
Nilpotent Lie Algebras ◽  
Simply Connected ◽  
Dense Set

AbstractDani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra 𝔫G from a simple directed graph G in 2005. There is a natural inner product on 𝔫G arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group N with Lie algebra 𝔫g. We classify singularity properties of the Lie algebra 𝔫g in terms of the graph G. A comprehensive description is given of graphs G which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph G and on a lattice Γ ⊆ N for which the quotient Γ \ N, a compact nilmanifold, has a dense set of smoothly closed geodesics. This paper provides the first investigation connecting graph theory, 2-step nilpotent Lie algebras, and the density of closed geodesics property.

scholarly journals Proper Actions of Certain Nilpotent Affine Groups with Codimension One Orbits

2012 ◽  
Vol 4 (2) ◽  
pp. 315
Author(s):  
N. Salma
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Groups ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Codimension One ◽  
Affine Groups ◽  
Simply Connected

Criterion for proper actions has been established for a homogeneous space of reductive type by Kobayashi (Math. Ann. 1989, 1996). On the other hand, an analogous criterion to Kobayashi’s equivalent conditions was proposed by Lipsman (1995) for a nilpotent Lie group . Lipsman's Conjecture: Let  be a simply connected nilpotent Lie group. Then the following two conditions on connected subgroups  and  are equivalent: (i) the action of  on  is proper; (ii)  is compact for any  The condition (i) is important in the study of discontinuous groups for the homogeneous space , while the second condition (ii) can easily be checked. The implication (i)  (ii) is obvious, and the opposite implication (ii)  (i) was known only in some lower dimensional cases. In this paper we prove the equivalence (i) ? (ii) for certain affine nilpotent Lie groups . Keywords: Affine nilpotent groups; Homogeneous manifolds; Proper actions; Properly discontinuous actions; Simply connected nilpotent Lie groups; Compact isotropy property (CI);  Eigenvalues. © 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v4i2.7889 J. Sci. Res. 4 (2), 315-326 (2012)

scholarly journals Parameter rigid actions of simply connected nilpotent Lie groups

2012 ◽  
Vol 33 (6) ◽  
pp. 1864-1875 ◽  
Author(s):  
HIROKAZU MARUHASHI
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Compact Manifold ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
Dense Orbit ◽  
Simply Connected

AbstractWe show that for a locally free $C^{\infty }$-action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real-valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys.27(2007), 1719–1735] concerning the Heisenberg groups.

scholarly journals The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}

2016 ◽  
Vol 28 (4) ◽  
Author(s):  
Giovanni Falcone ◽  
Ágota Figula
Keyword(s):  
Lie Algebras ◽  
Lie Group ◽  
Compact Lie Group ◽  
Nilpotent Lie Algebras ◽  
Group Of Automorphisms ◽  
Finite Dimensional

AbstractWe classify finite-dimensional real nilpotent Lie algebras with 2-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to

scholarly journals POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

2019 ◽  
Vol 62 (3) ◽  
pp. 706-736
Author(s):  
DAVID KYED ◽  
HENRIK DENSING PETERSEN
Keyword(s):  
Lie Group ◽  
Nilpotent Groups ◽  
Group Cohomology ◽  
Exponential Map ◽  
Nilpotent Lie Group ◽  
Polynomial Maps ◽  
Polynomial Coefficients ◽  
Classical Polynomials ◽  
Simply Connected ◽  
Refined Version

AbstractWe introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map.

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