2021/12/10 by Benjamin Barrett, David Hume, Barrett, Benjamin +3
Mathematics · Physics and Astronomy · #51F30 #53C23 #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2112.05305
openalex publication_date 2021/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 CNln(1+N) and prove that this is optimal. For rank at most 1 we prove lower bounds of the form cNa 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≥ 3 every bounded degree graph admits a coarse wiring into ℝk with volume at most CN1+(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.