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

Daniel N. Levitin

TL;DR

This work addresses the large-scale geometry of left-invariant distances on nondegenerate higher-rank Sol-type groups with perpendicular splittings, establishing that the rough similarity type is completely determined by the induced metric on the $\\mathbb{R}^k$ factor. It develops a robust framework based on half-space visiting box paths and Euclidean curve surgery to compare metrics, proving a two-sided distance inequality governed by top and bottom eigenvalues and identifying the metric moduli with the symmetric space $SL_k(\\mathbb{R})/SO_k(\\mathbb{R})$. A key technical contribution is the reduction of geometric questions to HSV geometry via the box-geodesic distance $\\rho_g$, shown to approximate the true Riemannian distance up to an additive constant. The results yield intrinsic rough-isometry information conditional on a Peng-type conjecture and provide a pathway toward a metric-structure classification for higher-rank Sol-type groups, with potential generalizations to broader solvable groups.

Abstract

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 $\textit{Euclidean curve surgery}$.

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

TL;DR

This work addresses the large-scale geometry of left-invariant distances on nondegenerate higher-rank Sol-type groups with perpendicular splittings, establishing that the rough similarity type is completely determined by the induced metric on the factor. It develops a robust framework based on half-space visiting box paths and Euclidean curve surgery to compare metrics, proving a two-sided distance inequality governed by top and bottom eigenvalues and identifying the metric moduli with the symmetric space . A key technical contribution is the reduction of geometric questions to HSV geometry via the box-geodesic distance , shown to approximate the true Riemannian distance up to an additive constant. The results yield intrinsic rough-isometry information conditional on a Peng-type conjecture and provide a pathway toward a metric-structure classification for higher-rank Sol-type groups, with potential generalizations to broader solvable groups.

Abstract

In this paper, we generalize the results of (, forthcoming) to describe the split left-invariant Riemannian distances on higher-rank Sol-type groups . We show that the rough isometry type of such a distance is determined by a specific restriction of the metric to , and therefore the space of rough similarity types of distances is parameterized by the symmetric space . In order to prove this result, we describe a family of uniformly roughly geodesic paths, which arise by way of the new technique of .

Paper Structure

This paper contains 8 sections, 25 theorems, 61 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie algebraic properties of splittings
  4. Geometric properties of splittings
  5. Comparing lengths of box geodesics
  6. Surgery in Euclidean Space
  7. Existence of half-space visiting coarse geodesics.
  8. Intrinsic rough isometries

Key Result

Theorem 1.0.1

Let $G=\prod_{i=1}^n \mathbf{N}_i\rtimes \mathbb{R}^k$ be a higher-rank Sol-type group. Let $g_1$ and $g_2$ be perpendicularly split left-invariant Riemannian metrics, and let their induced metrics on the $\mathbb{R}^k$ factors be equal (resp. similar). Then the related left-invariant Riemannian dis

Figures (5)

  • Figure 1: Schematic of the path surgery. On each of $n^2$ sub-intervals, we will preserve the $\vec{v}^{\perp}$-component of the path, and replace the $\vec{v}$ component of the path with a piecewise-linear function of the $\vec{v}^\perp$ arc length. If the curve is tangent to the $\vec{v}$-direction on some interval, projection down to $\vec{v}^\perp$ gets rid of this interval.
  • Figure 2: The cases of the proof are illustrated in this figure. The subcurves $I_1$ and $\mathscr{C}_1$ are both bounded by the indicated bold points. Either a curve like $\mathscr{C}_1$ must be long in coordinates perpendicular to $\nabla\alpha_i$, or else there must be at least $n^2$ many subcurves like $I_1$, for which reflecting the part below $H_i-r$ yields an equally-long curve that reaches height $H_i$.
  • Figure 3: Above: The curve $\mathscr{C}$ is the dashed multicurve. The interval $I_1$ goes from the beginning of the curve until the location of the loop surgery. The interval $I_2$ goes from the loop surgery until the beginning of the path surgery, including parts of $q(\gamma)$ which are not in $\mathscr{C}$, and $I_3$ runs from the end of the path surgery until the end of the curve. Below: performing surgery on the bolded locations, and then changing $q(\gamma(t))$ on the intervals $I_1$, $I_2$, and $I_3$ to match the dotted lines, yields the curve $\xi_1$. Since $\ell\circ q(\mathscr{C})$ is long, the smaller $\int_\mathscr{C} \| \frac{d(q(\gamma(t))}{dt}|_{\nabla\alpha_i^\perp}\| dt$ is, the larger the variation of $\alpha_i$ must be on $\mathscr{C}$, despite the fact that $\mathscr{C}$ has $\nabla\alpha_i$-coordinate between $b$ and $b+r$. If this variation is too great, then $\xi_1$ is shorter than the geodesic $\gamma$.
  • Figure 4: Slicing the geodesic $\gamma$ into pieces. $d_{i, H_i}$-distance is shown on the $x$-axis. One of the slices must traverse a sufficiently large fraction of the $d_{i, H_i}$ distance far enough below the half-space $H_i$ that we can use Lemma \ref{['ObtainingLongPerpendicularLength']}.
  • Figure 5: Determining surgeries for the curve $\xi_1$. The cross-section is the same as before, but bolded are surgery locations of options of the first surgery family. As long as there are enough options for each family, we can choose options that do not cover much of the desired slice of the curve. For instance, supposing that $j_2=1$, we could choose the rightmost surgery option.

Theorems & Definitions (63)

  • Theorem 1.0.1
  • Theorem 1.0.2
  • Theorem 1.0.3
  • Theorem 1.0.4
  • Definition 2.0.1
  • Definition 2.0.2
  • Definition 2.0.3
  • Definition 2.0.4
  • Definition 2.0.5
  • Definition 2.0.6
  • ...and 53 more