Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Thick embeddings of graphs into symmetric spaces via coarse geometry
HTML articles powered by AMS MathViewer

by Benjamin Barrett, David Hume, Larry Guth and Elia Portnoy;
Trans. Amer. Math. Soc. 378 (2025), 885-909
DOI: https://doi.org/10.1090/tran/9109
Published electronically: December 12, 2024

Abstract:

We prove estimates for the optimal volume of thick embeddings of finite graphs into symmetric spaces, generalising results of Kolmogorov-Barzdin and Gromov-Guth for embeddings into Euclidean spaces. We distinguish two very different behaviours depending on the rank of the non-compact factor. For rank at least 2, we construct thick embeddings of $N$-vertex graphs with volume $CN\ln (1+N)$ and prove that this is optimal. For rank at most $1$ we prove lower bounds of the form $cN^a$ for some (explicit) $a>1$ which depends on the dimension of the Euclidean factor and the conformal dimension of the boundary of the non-compact factor. The main tool is a coarse geometric analogue of a thick embedding called a coarse wiring, with the key property that the minimal volume of a thick embedding is comparable to the “minimal volume” of a coarse wiring for symmetric spaces of dimension at least $3$. In the appendix it is proved that for each $k\geq 3$ every bounded degree graph admits a coarse wiring into $\mathbb {R}^k$ with volume at most $CN^{1+\frac {1}{k-1}}$. As a corollary, the same upper bound holds for real hyperbolic space of dimension $k+1$ and in both cases this result is optimal.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 51F30, 53C23
  • Retrieve articles in all journals with MSC (2020): 51F30, 53C23
Bibliographic Information
  • Benjamin Barrett
  • MR Author ID: 1279644
  • David Hume
  • MR Author ID: 1029452
  • ORCID: 0000-0003-2195-6071
  • Larry Guth
  • MR Author ID: 786046
  • Received by editor(s): April 19, 2022
  • Received by editor(s) in revised form: January 24, 2023, and November 28, 2023
  • Published electronically: December 12, 2024
  • Additional Notes: The appendix was written by the third and fourth authors
  • © Copyright 2024 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 885-909
  • MSC (2020): Primary 51F30; Secondary 53C23
  • DOI: https://doi.org/10.1090/tran/9109