Short incompressible graphs and 2-free groups (Q6620353)

Given a geodesic metric space \(X\), a basic question is to determine the systole of \(X\), i.e., the length of the shortest simple closed curve in \(X\) that cannot be homotoped to a point. This problem can be generalised to finding the shortest combined length of a finite collection of loops in \(X\), however for this to be interesting one has to impose some conditions on the loops. A natural choice is to require that the loops have a common basepoint and be independent, meaning that their images in \(\pi_1(X)\) generate a free group.\N\N\textit{M. Gromov} [J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037), Theorem 5.4.A] showed that if \(\Sigma\) is a Riemannian surface with negative Euler characteristic, then the length of the shortest pair of curves meeting the conditions above is at most \(2\sqrt{\mathrm{Area}(\Sigma)}\).\N\NIn the article under review, the authors generalise Gromov's result in the following way. Let \(X\) be a compact \(2\)-complex with a piecewise Riemannian metric (i.e., each cell is endowed with a Riemannian metric). Moreover, suppose that \(\pi_1(X)\) is freely indecomposable, not generated by fewer than \(3\) elements, and is \(2\)-free, meaning that all its subgroups generated by two elements are free (all of these conditions are met when \(X\) is a surface). Then the authors prove that there is an embedded graph \(\Gamma \subseteq X\) such that \(\pi_1(\Gamma) \cong F_2 \hookrightarrow \pi_1(X)\) and whose length is at most \(4\sqrt{2\mathrm{Area}(X)}\). The result is then used to show that the volume entropy of \(X\) is bounded away from zero.