We investigate symplectic nilpotent Lie groups with Lagrangian normal subgroups. We show that there exists a bijection between the isomorphism classes of nilpotent Lie groups with Lagrangian normal subgroups and the isomorphism classes of geodesically complete, flat, nilpotent Lie groups with Lagrangian extension cohomology class. Finally, we provide a complete classification of eight-dimensional symplectic nilpotent Lie groups with Lagrangian normal subgroups, identifying exactly ninety-five such groups. As a consequence, we obtain a complete classification of eight-dimensional symplectic filiform real Lie groups.

Abstract Around 1980 commutator theory was generalized from groups to arbitrary algebras using the socalled term condition commutator. The semigroups that are abelian with respect to this commutator were classified by Warne [22]. We study what solvability, nilpotence, and supernilpotence in the sense of commutator theory mean for semigroups and how these notions relate to classical concepts in semigroup theory. We show that a semigroup with a completely simple ideal is solvable (left nilpotent or right nilpotent or supernilpotent) in the sense of commutator theory iff it is a nilpotent extension in the classical sense of semigroup theory of a completely simple semigroup with solvable (nilpotent) subgroups. These characterizations hold in particular for finite semigroups and for eventually regular semigroups, i.e., semigroups in which every element has some regular power. We also show that a monoid is (left and right) nilpotent in the sense of commutator theory iff it embeds into a nilpotent group.

We study directed weighted graphs which are invariant under a nilpotent and cocompact group action. In particular, we consider the conic section K \mathcal {K} of the set of positive harmonic functions. We characterise the set of extreme points of the convex and compact set K \mathcal {K} as the set of multiplicative elements in K \mathcal {K} . Moreover, we study positive generalised eigenfunctions for a given parameter λ \lambda . We find that the topological space M λ \mathcal {M}_{\lambda } of multiplicative λ \lambda -harmonic functions is homeomorphic to a sphere for λ \lambda below a certain threshold.

Mekler's construction is a powerful technique for building purely algebraic structures from combinatorial ones.Its power lies in the fact that it allows various model-theoretic tameness properties of the combinatorial structure to transfer to the algebraic one.In this paper, we push this ideology much further, describing a broad class of properties that transfer through Mekler's construction.This technique subsumes many well-known results and opens avenues for many more.As a straightforward application of our methods, we obtain transfer principles for stably embedded pairs of Mekler groups and construct strictly NFOP k pure groups for all k >2 .We also answer a question of Chernikov and Hempel on transfer of burden.

In the present paper, taking a dynamical point on view, we study the growth rate and asymptotic behavior of the sequences of the Reidemeister numbers and the sequences of the Reidemeister and the Nielsen coincidence numbers. We also prove the Gauss congruences for the sequence $\{R(\varphi^n,\psi^n)\}$ of the Reidemeister coincidence numbers of the tame pair $(\varphi,\psi)$ of endomorphisms of a torsion-free nilpotent group $G$ of finite Pr\"ufer rank. Furthermore, we prove the rationality of the Nielsen coincidence zeta function, the Gauss congruences for the sequence $\{N(f^n, g^n)\}$ of the Nielsen coincidence numbers and show that the growth rate exists for the sequence \{$N(f^n, g^n)\}$ of tame pair of maps $(f,g)$ of a compact nilmanifold to itself.

For every algebraically closed field k k and natural number r r , we construct several algebraic varieties (over k k ) whose birational automorphism group contains every finite nilpotent group of class at most 2 2 , rank at most r r whose order is coprime to the characteristic of k k . This construction is sharp in characteristic 0 0 , i.e. up to bounded extension, the set of groups from the statement cannot be replaced by a larger one. Using similar main ideas (with different technical details), for every r r , we construct several compact manifolds whose diffeomorphism groups contain every finite nilpotent group of class at most 2 2 , rank at most r r . This result answers a question of Mundet i Riera affirmatively and is conjecturally sharp up to bounded extension.

In this work, we systematically treat the ambiguities that generically arise in the gradient expansion of any hydrodynamic theory. While these ambiguities do not affect the physical content of the equations, they induce two types of transformations in the space of transport coefficients. The first type is known as the “frame” transformations, and amounts to field redefinitions. The second type, which we introduce and formalize here, we term the “on-shell” transformations. This identifies equivalence classes of hydrodynamic theories that provide an equally valid low-energy description of the underlying microscopic theory. We show that in any (classical) theory of hydrodynamics (at arbitrary order in derivatives), the action of such transformations on the dispersion relations and two-point correlation functions is universal. We explicitly construct invariants which can then be matched to a microscopic theory. Among them are, expectedly, the low-momentum expansions of the hydrodynamic modes. The (unphysical) gapped modes can, however, be added or removed at will. Finally, we show that such transformations assign a nilpotent Lie group to every hydrodynamic theory, and discuss the related algebraic properties underlying classical hydrodynamics.

Abstract We establish upper bounds on the lengths of minimal conjugators in 2‐step nilpotent groups. These bounds exploit the existence of small integral solutions to systems of linear Diophantine equations. We prove that in some cases these bounds are sharp. This enables us to construct a family of finitely generated 2‐step nilpotent groups such that the conjugator length function of grows like a polynomial of degree .

Abstract We study the positive Hermitian curvature flow for left-invariant metrics on 2-step nilpotent Lie groups G with a left-invariant complex structure J . We describe the long-time behavior of the flow under the assumption that $$J[\mathfrak {g}, \mathfrak {g}]$$ J [ g , g ] is contained in the center of $$\mathfrak {g}$$ g . We show that under our assumption the flow $$g_{t}$$ g t exists for all positive t and $$(G,(1+t)^{-1}g_{t})$$ ( G , ( 1 + t ) - 1 g t ) converges, in the Cheeger-Gromov topology, to a 2-step nilpotent Lie group with a non flat semi-algebraic soliton. Moreover, we prove that, in our class of Lie groups, there exists at most one semi-algebraic soliton solution, up to homothety. Similar results were proved by M. Pujia and J. Stanfield for nilpotent complex Lie groups [21, 24]. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.

We give a description of the Linnell division ring of a countable residually (poly- Z \mathbb {Z} virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group G G with coefficients in this Novikov ring implies the existence of a normal subgroup N ⩽ G N \leqslant G such that c d Q ( N ) > c d Q ( G ) cd_\mathbb {Q}(N) > cd_\mathbb {Q}(G) and G / N G/N is poly- Z \mathbb {Z} virtually nilpotent. As a consequence, we show that if G G is an RPVN group of finite type, then its top-degree ℓ 2 \ell ^2 -Betti number vanishes if and only if there is a poly- Z \mathbb {Z} virtually nilpotent quotient G / N G/N such that c d Q ( N ) > c d Q ( G ) cd_\mathbb {Q}(N) > cd_\mathbb {Q}(G) . In particular, finitely generated RPVN groups of cohomological dimension 2 2 are virtually free-by-nilpotent if and only if their second ℓ 2 \ell ^2 -Betti number vanishes, and therefore 2 2 -dimensional RPVN groups with vanishing second ℓ 2 \ell ^2 -Betti number are coherent. As another application, we show that if G G is a finitely generated parafree group with c d ( G ) = 2 cd(G) = 2 , then G G satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.

Consider the random Cayley graph of a finite group G with respect to k generators chosen uniformly at random, with 1 \ll \log k \ll{\log}|G| ; denote it by G_{k} . A conjecture of Aldous and Diaconis (1985) asserts, for k \gg{\log}|G| , that the random walk on this graph exhibits cutoff. Further, the cutoff time should be a function only of k and |G| , to sub-leading order. This was verified for all Abelian groups in the ’90s. We extend the conjecture to 1 \ll k \lesssim{\log}|G| . We establish cutoff for all Abelian groups under the condition k - d(G) \gg 1 , where d(G) is the minimal size of a generating subset of G , which is almost optimal. The cutoff time is described (abstractly) in terms of the entropy of random walk on \mathbb{Z}^{k} . This abstract definition allows us to deduce that the cutoff time can be written as a function only of k and |G| when d(G) \ll{\log}|G| and k - d(G) \asymp k \gg 1 ; this is not the case when d(G) \asymp{\log}|G| \asymp k . For certain regimes of k , we find the limit profile of the convergence to equilibrium. Wilson (1997) conjectured that \mathbb{Z}_{2}^{d} gives rise to the slowest mixing time for G_{k} amongst all groups of size at most 2^{d} . We give a partial answer, verifying the conjecture for nilpotent groups. This is obtained via a comparison result of independent interest between the mixing times of nilpotent G and a corresponding Abelian group \overline G , namely the direct sum of the Abelian quotients in the lower central series of G . We use this to refine a celebrated result of Alon and Roichman 1994: we show for nilpotent G that G_{k} is an expander provided k - d(\overline G) \gtrsim{\log}|G| . As another consequence, we establish cutoff for nilpotent groups with relatively small commutator subgroup, including high-dimensional special groups, such as Heisenberg groups. The aforementioned results all hold with high probability over the random Cayley graph G_{k} .

Abstract Approximate lattices are a class of approximate subgroups (i.e. subsets of groups closed under multiplication up to a finite error) that generalise lattices of locally compact groups. We provide and motivate in this paper a natural framework for the study of approximate lattices. Namely, we consider approximate lattices in so-called S -adic linear groups and define relevant notions of arithmeticity involving Pisot numbers. We also adapt to this framework classical results of the theory of lattices and Meyer sets. Results from this paper will play a role in the proof of a structure theorem for approximate lattices in S -adic linear groups which is the subject of a companion paper. We extend a theorem of Schreiber’s concerning the coarse structure of approximate subgroups in Euclidean spaces to approximate subgroups of unipotent S -adic groups. We generalise Meyer’s structure theorem for approximate lattices in locally compact abelian groups to a precise structure theorem for approximate lattices in unipotent S -adic groups. Finally, we study intersections of approximate lattices of S -adic linear groups with certain subgroups such as the nilpotent radical and Levi subgroups in the spirit of a theorem of Bieberbach. We furthermore show that the framework of S -adic linear groups enables us to provide statements more precise than earlier results.

Abstract We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact‐by‐discrete. Notable examples include finitely generated nilpotent groups, most lattices in semisimple Lie groups, and irreducible nongeometric 3‐manifold groups. We show graphs of groups with graphically discrete vertex groups frequently have strong rigidity properties. We prove free products of one‐ended virtually torsion‐free graphically discrete groups are action rigid within the class of virtually torsion‐free groups. We also prove quasi‐isometric rigidity for many hyperbolic graphs of groups whose vertex groups are closed hyperbolic manifold groups and whose edge groups are nonelementary quasi‐convex subgroups. This includes the case of two hyperbolic 3‐manifold groups amalgamated along a quasi‐convex malnormal non‐abelian free subgroup. We provide several additional examples of graphically discrete groups and illustrate this property is not a commensurability invariant.

This paper continues Representation zeta function of a family of maximal class groups: non-exceptional primes [7]. Using the explicit methods from the prequel, we compute exceptional cases of p-local representation zeta functions for finitely generated nilpotent groups M n of maximal class. Specifically, we construct all irreducible representations of dimension p N for M p + 1 and dimension 2 N for M 4 . Combined with previous results, this completes the classification of irreducible representations for M 3 and M 4 and their global representation zeta functions.

On A-groups with the same index set as a nilpotent group

Gradings on nilpotent Lie algebras associated with the nilpotent fundamental groups of smooth complex algebraic varieties

Let [Formula: see text] be a division ring with center [Formula: see text] and multiplicative group [Formula: see text], of char [Formula: see text], and with an involution *. Let [Formula: see text] be the group of unitary units of [Formula: see text], namely [Formula: see text]. We investigate various instances where the dimension [Formula: see text], and in which every non-central normal subgroup [Formula: see text] contains a free non cyclic subgroup. Among them, we consider the cases where [Formula: see text] is either generated over its center of characteristic 0 by a torsion free nilpotent group, or [Formula: see text] is the field of fractions of a group algebra [Formula: see text] of the residually torsion free nilpotent group [Formula: see text] over the field [Formula: see text] of characteristic 0 , or the field of fractions of the enveloping algebra of a locally solvable residually nilpotent Lie [Formula: see text]-algebra [Formula: see text].

Motivated by a conjecture formulated by I. Shestakov, we investigate the local finiteness of coalgebras in certain varieties of noncommutative Jordan algebras that admit the locally nilpotent radical. It is proved that if a variety is admissible (in the sense of [12]), then the respective coalgebras are locally finite. It is still an open problem to determine if coalgebras belonging to locally admissible varieties (in the sense of [12]) are also locally finite, but we present an example of variety that is locally admissible and not admissible, such that its coalgebras are locally finite. We also show an example of a variety of noncommutative Jordan algebras such that the respective coalgebras are locally finite, but the question of existence of the locally nilpotent radical is still open.

Let [Formula: see text] be a group and [Formula: see text] the group of all central automorphisms of [Formula: see text]. Denote by [Formula: see text] the group of all central automorphisms of [Formula: see text] fixing [Formula: see text] elementwise. We characterize all nilpotent groups [Formula: see text] of class 2 with the minimal condition on subgroups for which [Formula: see text]. Let [Formula: see text] be a central product of a finitely generated nilpotent group and a nilpotent group with the minimal condition on subgroups. When the nilpotency class of [Formula: see text] is 2, we characterize [Formula: see text] for which [Formula: see text].

The Cohen–Lenstra–Martinet heuristics lead one to conjecture that the average size of the p -torsion in class groups of G -extensions of a number field is finite. In a 2021 paper, Lemke Oliver, Wang, and Wood proved this conjecture in the case of p = 3 for permutation groups G of the form C 2 ≀ H for a broad family of permutation groups H , including most nilpotent groups. However, their theorem does not apply for some nilpotent groups of interest, such as H = C 5 . We extend their results to prove that the average size of 3 -torsion in class groups of C 2 ≀ H -extensions is finite for any nilpotent group H .