Groups acting on horocyclic products

TL;DR

The paper analyzes finitely generated groups acting geometrically on horocyclic products $X\bowtie Y$ of proper, geodesically complete $\mathop{CAT}(-\kappa)$ spaces, showing a boundary-driven trichotomy: either the group is not finitely presented (if both boundary components are disconnected), an ascending HNN extension of a finitely generated virtually nilpotent group (if exactly one boundary component is connected), or a finite-index subgroup that is a semidirect product of a virtually nilpotent group with $\mathbb{Z}$ (if both boundaries are connected). The authors develop a new metric $d'_X$ on $X$ and analyze the isometry group of horocyclic products by decomposing actions into factorwise components on $X'$ and $Y'$, proving a near-product structure up to a $\mathbb{Z}/2\mathbb{Z}$ swap. A key technique is upgrading arbitrary geometric actions to actions on millefeuille spaces $Z\bowtie W$ via Caprace–Cornulier–Monod–Tessera, enabling a boundary-to-algebra correspondence that detects Heintze versus nilpotent factors and yields strong structural conclusions about the acting groups. The results unify and extend known examples (lamplighter, BS(1,n), and $\mathbb{Z}^2\rtimes_A\mathbb{Z}$ actions) and provide a pathway toward quasi-isometric rigidity insights for horocyclic products. The work leverages Ferragut’s visual-boundary framework and CCMT’s Millefeuille theory to connect geometric properties of boundaries with exact algebraic decompositions.

Abstract

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 $X\bowtie Y$.

Figures (11)

• Figure 1: The horocyclic product $\mathbb{H}^2\bowtie T_2$ (left) and the millefeuille space $\mathbb{H}^2[2]$ (right) with dotted geodesics. Both are trees of hyperbolic planes, but the $\mathbb{H}^2$-leaves are embedded with opposite heights.
• Figure 2: The horocyclic product of two trees might itself be a tree if we choose the height function badly.
• Figure 3: Illustration of the proof of Lemma \ref{['lemma:BoundedCoordinateDinstancesImpliesBoundedDistance']}. The first, second, and third subpaths are dashed, dotted, and solid respectively. $y_3$ is a point on $V_{Y, y_1}$ at height $-h_X(x_2)$, and $x_3$ is a point on $V_{X, x_2}$ at height $-h_Y(y_1)$
• Figure 4: Illustration of the proof of Lemma \ref{['lemma:CAT-1ImpliesBusemann']}. Triangles $\triangle X_1X_2X_3$ and $\triangle X_2X_3X_4$ are comparison triangles for $\triangle x_1x_2x_3$ and $\triangle x_2x_3x_4$ respectively.
• Figure 5: Illustration of the proof of Lemma \ref{['lemma:CAT-1ImpliesVerticalConvergence']}. On the left are points in $X$, and on the right are points in $\mathbb{H}^2$. The point $X_{2,T}$ varies along the righthand circle.

Theorems & Definitions (100)

• Definition 1.0.1: Heintze group
• Definition 1.0.2: Millefeuille space
• Definition 2.1.1
• Definition 2.1.2
• Definition 2.1.3
• Definition 2.1.4
• Lemma 2.1.5
• Definition 2.1.6
• Definition 2.1.7
• Definition 2.1.8