arXiv subjects

Daniel N. Levitin

Publications and source records attributed to Daniel N. Levitin.

4 recordsLinked to original sources

arXiv2025

Groups acting on horocyclic products

Horocyclic products are a well-studied class of metric spaces that provide models for various solvable Lie groups, Baumslag-Solitar groups, and Lamplighter groups. Let $G$ act geometrically on a horocyclic product $X \bowtie Y$ of $\CAT(-κ)$ spaces $X,Y$. We show that every such group is either an ascending HNN extension of a finitely-generated virtually nilpotent group, or else is not finitely presented, depending on the connectivity of the visual boundary of $X\bowtie Y$.

math.GR↗

arXiv2025

Left-Invariant Riemannian Distances on Higher-Rank Sol-Type Groups

In this paper, we generalize the results of ($\textit{Groups, Geom. Dyn.}$, forthcoming) to describe the split left-invariant Riemannian distances on higher-rank Sol-type groups $G=\mathbf{N}\rtimes \mathbb{R}^k$. We show that the rough isometry type of such a distance is determined by a specific restriction of the metric to $\mathbb{R}^k$, and therefore the space of rough similarity types of distances is parameterized by the symmetric space $SL_k(\mathbb{R})/SO_k(\mathbb{R})$. In order to prove this result, we describe a family of uniformly roughly geodesic paths, which arise by way of the new technique of $\textit{Euclidean curve surgery}$.

math.GR↗

arXiv2024

Finite-State Machines for Horospheres in Hyperbolic Right-Angled Coxeter Groups

Relatively little is known about the discrete horospheres in hyperbolic groups, even in simple settings. In this paper we work with hyperbolic one-ended right-angled Coxeter groups and describe two graph structures that mimic the intrinsic metric on a classical horosphere: the Rips graph and the divergence graph (the latter due to Cohen, Goodman-Strauss, and Rieck). We develop, analyze, and implement algorithms based on finite-state machines that draw large finite portions of these graphs, and deduce various geometric corollaries about the path metrics induced by these graph structures.

math.MG↗

arXiv2023

Metric Spaces of Arbitrary Finitely-Generated Scaling Group

For a metric space $X$ with a compatible measure $μ$, Genevois and Tessera defined the Scaling Group of $(X,μ)$ as the subgroup $Γ$ of $\mathbb{R}_{>0}$ of positive real numbers $γ$ for which there are quasi-isometries of $X$ coarsely scaling $μ$ by a factor of $γ$. We show that for any finitely generated subgroup $Γ$ of $\mathbb{R}_{>0}$ there exists a space $N_Γ$, bi-Lipschitz equivalent to a graph of finite degree, with scaling group $Γ$.

math.MG↗