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Linear Diophantine equations and conjugator length in 2-step nilpotent groups
Martin R. Bridson, Timothy R. Riley
https://arxiv.org/abs/2506.01239 https://…

Linear Diophantine equations and conjugator length in 2-step nilpotent groups
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 $(G_m)_{m\in\mathbb{N}}$ such that the conjugator length function of $G_m$ grows like a polynomial of degree $m+1$.

This https://arxiv.org/abs/2406.06819 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Classification of almost abelian Lie groups admitting left-invariant complex or symplectic structures
We classify the almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0 \ltimes_A \mathbb R^{2n-1}$ admitting complex or symplectic structures. The matrix $A\in M(2n-1,\mathbb R )$ encodes the adjoint action of $e_0$ on the abelian ideal $\mathbb R^{2n-1}$, and the existence of complex or symplectic structures on $\mathfrak g_A$ imposes restrictions on the Jordan normal form of $A$. The classification essentially reduces to the case when $A$ is nilpotent, so we start by considering this case. …

This https://arxiv.org/abs/2501.03636 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the Nowicki Conjecture for the free Lie algebra of rank 2
Let K[X_n]=K[x_1,\ldots,x_n] be the polynomial algebra in n variables over a field K of characteristic zero. A locally nilpotent linear derivation δof K[X_n] is called Weitzenböck due to his well known result from 1932 stating that the algebra \text{\rm ker}(δ)=K[X_n]^δ of constants of $δ$ is finitely generated. The explicit form of a generating set of $K[X_n,Y_n]^δ$ was conjectured by Nowicki in 1994 in the case δwas such that δ(y_{i})=x_{i}$, $δ(x_i)=0, i=1,\ldots,n. Nowicki's conjec…

This https://arxiv.org/abs/2505.13260 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Some remarks on Quillen's Dévissage theorem
In this paper we give a different proof of Quillen's Dévissage theorem using Barwick's theorem of the heart. The key ingredient is a certain short exact sequence of dg categories, which is closely related with the Auslander-type construction for nilpotent extensions which was used in the papers of Kuznetsov-Lunts \cite{KL15}, Land-Tamme \cite{LT19} and the author \cite{E20}.

This https://arxiv.org/abs/2304.07852 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Simplicity of crossed products by FC-hypercentral groups
In this paper, we give a complete, two-way characterization, of when a noncommutative crossed product $A \rtimes_λG$ is simple, in the case of $G$ being an FC-hypercentral group. This is a large class of amenable groups that contains all virtually nilpotent groups, and in the finitely-generated setting, coincides with the set of groups which have polynomial growth. We further completely characterize the ideal intersection property under the assumption that the group is FC, meaning that every e…

This https://arxiv.org/abs/2503.05572 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Word problems and embedding-obstructions in cellular automata groups on groups
We study groups of reversible cellular automata, or CA groups, on groups. More generally, we consider automorphism groups of subshifts of finite type on groups. It is known that word problems of CA groups on virtually nilpotent groups are in co-NP, and can be co-NP-hard. We show that under the Gap Conjecture of Grigorchuk, their word problems are PSPACE-hard on all other groups. On free and surface groups, we show that they are indeed always in PSPACE. On a group with co-NEXPTIME word problem, …