2023/04/21 by Florent Balacheff, Balacheff, Florent, Wolfgang Pitsch +1
Mathematics · #20F34 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Primary: 53C23 #Secondary: 20F05
paper · pdf · doi:10.48550/arxiv.2304.10924
openalex publication_date 2023/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Consider a finite connected 2-complex X endowed with a piecewise Riemannian metric and whose fundamental group is freely indecomposable, of rank at least 3, and in which every 2-generated subgroup is free. In this paper we show that we can always find a connected graph Γ⊂ X such that π1 Γ≃ \mathbb F2 \hookrightarrowπ1 X (in short, a 2-incompressible graph) whose length satisfies the following curvature-free inequality: ℓ(Γ)≤ 4√(2Area(X)). This generalizes a previous inequality proved by Gromov for closed Riemannian surfaces with negative Euler characteristic. As a consequence we obtain that the volume entropy of such 2-complexes with unit area is always bounded away from zero.