arXiv subjects

J. Maxwell Riestenberg

Publications and source records attributed to J. Maxwell Riestenberg.

10 recordsLinked to original sources

arXiv2026

Harmonic maps to Hadamard spaces and a universal higher Teichmüller space

We give a sufficient criterion, which we call stability, for a coarse Lipschitz map $f$ from a complete manifold $X$ with Ricci curvature bounded below to a proper Hadamard space $Y$ to be within bounded distance of a harmonic map. We prove uniqueness of the harmonic map under additional assumptions on $X$ and $Y$. Using this criterion, we prove a significant generalization of the Schoen-Li-Wang conjecture on quasi-isometric embeddings between rank 1 symmetric spaces. In particular, under a natural generalization of the quasi-isometric condition, we remove the assumption that the target has rank 1. This allows us to define a universal Hitchin component for each $\mathrm{PGL}_d(\mathbb{R})$, generalizing universal Teichmüller space, and show that it can be described both as a space of quasi-symmetric positive maps from $\mathbb{RP}^1$ to the flag variety, and as a space of harmonic maps.

math.DG↗

arXiv2026

Certifying Arithmeticity for Two Degree-Six Symplectic Hypergeometric Monodromy Groups

We prove arithmeticity for two degree-six symplectic hypergeometric monodromy groups, called C-47, and C-55 in the paper \cite{BajpaiDonaNitsche2025Thin} by Bajpai-Dona-Nitsche. This settles two of the three remaining cases, whose classification was left open by \cite{BajpaiDonaNitsche2025Thin}. The arithmeticity certificates were found with AlphaEvolve and then independently verified with exact matrix arithmetic over $\QQ$ using a computer. We include illustrations of limit sets of several degree-six symplectic hypergeometric monodromy groups. Based on these illustration we conjecture C-32 to be thin.

math.GR↗

arXiv2026

Dirichlet domains for Anosov subgroups

We introduce a sufficient condition for a finitely generated subgroup $Γ$ of a semisimple Lie group $G$ to admit finite-sided Dirichlet domains for polyhedral Finsler metrics on the symmetric space $G/K$. The condition always implies the $Θ$-Anosov condition for some $Θ$, and can be arranged to be equivalent to the $Θ$-Anosov condition when $G$ is simple and $Θ$ is the set of long roots or the set of short roots. The Dirichlet domain we obtain extends to a fundamental domain for the action of $Γ$ on a domain of discontinuity in a flag manifold. For instance, Borel Anosov subgroups of $\mathrm{SL}(d,\mathbb{R})$ have finite-sided Dirichlet domains for the Hilbert metric on the symmetric space which extends to the space of line-hyperplane flags, and $n$-Anosov subgroups of $\mathrm{Sp}(2n,\mathbb{R})$ have finite-sided Dirichlet-Selberg domains in $\mathrm{SL}(2n,\mathbb{R})/\mathrm{SO}(2n)$ which extend to a domain in projective space bounded by quadrics.

math.GT↗

arXiv2026

Certifying Anosov representations

By providing new finite criteria which certify that a finitely generated subgroup of $\mathrm{SL}(d,\mathbb{R})$ or $\mathrm{SL}(d,\mathbb{C})$ is projective Anosov, we obtain a practical algorithm to verify the Anosov condition. We demonstrate on a surface group of genus 2 in $\mathrm{SL}(3,\mathbb{R})$ by verifying the criteria for all words of length 8. The previous version required checking all words of length $2$ million.

math.GR↗

arXiv2025

Transverse Spheres in Flag Manifolds

For some partial flag manifolds of semisimple real Lie groups, including many full flag manifolds, transverse circles are known to be locally maximally transverse. We complete the classification of all partial flag manifolds of split real Lie groups with this property. As a consequence, $\{7\}$-Anosov subgroups of split $E_7$ are virtually free or surface groups. On the other hand, using spinors, we find transverse spheres of arbitrarily large dimension in certain full flag manifolds of Cartan-Killing types $A,B,D$. These transverse spheres are verified to be maximally transverse with tools from topological $K$-theory. The aforementioned classification follows from constructions of transverse $m$-spheres, $m \geq 2$, that complement the previously known restrictions as well as the new $E_7$ restriction. Additionally, when $G$ is split of type $G_2, B_3,$ or $D_4$, the full flag manifold admits a fibration by maximally transverse 3-spheres.

math.GT↗

arXiv2024

Concave Foliated Flag Structures and the $\text{SL}_3(\mathbb{R})$ Hitchin Component

We give a geometric characterization of flag geometries associated to Hitchin representations in $\text{SL}_3(\mathbb{R})$. Our characterization is based on distinguished invariant foliations, similar to those studied by Guichard-Wienhard in $\text{PSL}_4(\mathbb{R})$. We connect to the dynamics of Hitchin representations by constructing refraction flows for all positive roots in general $\mathfrak{sl}_n(\mathbb{R})$ in our setting. For $n = 3$, leaves of our one-dimensional foliations are flow-lines. One consequence is that the highest root flows are $C^{1+α}$.

math.GT↗

arXiv2023

Normed Spaces for Graph Embedding

Theoretical results from discrete geometry suggest that normed spaces can abstractly embed finite metric spaces with surprisingly low theoretical bounds on distortion in low dimensions. In this paper, inspired by this theoretical insight, we highlight normed spaces as a more flexible and computationally efficient alternative to several popular Riemannian manifolds for learning graph embeddings. Normed space embeddings significantly outperform several popular manifolds on a large range of synthetic and real-world graph reconstruction benchmark datasets while requiring significantly fewer computational resources. We also empirically verify the superiority of normed space embeddings on growing families of graphs associated with negative, zero, and positive curvature, further reinforcing the flexibility of normed spaces in capturing diverse graph structures as graph sizes increase. Lastly, we demonstrate the utility of normed space embeddings on two applied graph embedding tasks, namely, link prediction and recommender systems. Our work highlights the potential of normed spaces for geometric graph representation learning, raises new research questions, and offers a valuable tool for experimental mathematics in the field of finite metric space embeddings. We make our code and data publically available.

cs.LG↗

arXiv2023

Restrictions on Anosov subgroups of Sp(2n,R)

Let $n\in\mathbb{N}$ and let $Θ\subset \{1,\dots,n\}$ be a non-empty subset. We prove that if $Θ$ contains an odd integer, then any $P_Θ$-Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ is virtually isomorphic to a free group or a surface group. In particular, any Borel Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ is virtually isomorphic to a free or surface group. On the other hand, if $Θ$ does not contain any odd integers, then there exists a $P_Θ$-Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ which is not virtually isomorphic to a free or surface group. We also exhibit new examples of maximally antipodal subsets of certain flag manifolds; these arise as limit sets of rank $1$ subgroups.

math.GT↗

arXiv2023

Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices

Recent research has shown that alignment between the structure of graph data and the geometry of an embedding space is crucial for learning high-quality representations of the data. The uniform geometry of Euclidean and hyperbolic spaces allows for representing graphs with uniform geometric and topological features, such as grids and hierarchies, with minimal distortion. However, real-world graph data is characterized by multiple types of geometric and topological features, necessitating more sophisticated geometric embedding spaces. In this work, we utilize the Riemannian symmetric space of symmetric positive definite matrices (SPD) to construct graph neural networks that can robustly handle complex graphs. To do this, we develop an innovative library that leverages the SPD gyrocalculus tools \cite{lopez2021gyroSPD} to implement the building blocks of five popular graph neural networks in SPD. Experimental results demonstrate that our graph neural networks in SPD substantially outperform their counterparts in Euclidean and hyperbolic spaces, as well as the Cartesian product thereof, on complex graphs for node and graph classification tasks. We release the library and datasets at \url{https://github.com/andyweizhao/SPD4GNNs}.

cs.LG↗

arXiv2021

A quantified local-to-global principle for Morse quasigeodesics

In arXiv:1403.7671, Kapovich, Leeb and Porti gave several new characterizations of Anosov representations $Γ\to G$, including one where geodesics in the word hyperbolic group $Γ$ map to "Morse quasigeodesics" in the associated symmetric space $G/K$. In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics and describe an algorithm which can verify the Anosov property of a given representation in finite time. However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. In this paper, we supplement their work with estimates in the symmetric space to obtain the first explicit criteria for their local-to-global principle. This makes their algorithm for verifying the Anosov property effective. As an application, we demonstrate how to compute explicit perturbation neighborhoods of Anosov representations with two examples.

math.DG↗