Systolic invariants of groups and 2-complexes via Grushko decomposition

Systoles of 2-complexes, Reeb graph, and Grushko decomposition

Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov's inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko's decomposition of the fundamental group. More specifically, we construct a kind of a Reeb space ``minimal m…

Relative systoles of relative-essential 2-complexes

We prove a systolic inequality for the phi-relative 1-systole of a phi-essential 2-complex, where phi is a homomorphism from the fundamental group of the complex, to a finitely presented group G. Indeed we show that universally for any phi-essential Riemannian 2-complex, and any G, the area of X is bounded below by 1/8 of sys(X,phi)^2. Combining our results with a method of Larry Guth, we obtain new quantitative results for certain 3-manifolds: in particular for Sigma the Poincare homology sphere, we have sys(Sigma)^3 < 24 vol(Sigma).

小分离器、l^∞-宽度的上界与 systole 几何

Small separators, upper bounds for $l^\infty$-widths, and systolic geometry

该研究探讨闭子流形的欧氏体积与$l^\infty$-宽度对维度的依赖，推导两类$l^\infty$-宽度上界，作为推论得到本质子流形的非可缩闭曲线存在性，还引入了 systole 几何的非线性新方法。