A systolic inequality for 2-complexes of maximal cup-length and systolic area of groups (Q6564132)

The systole sys\((X)\) of a metric space \(X\) is a metric invariant of \(X\) which is the length of a shortest non-contractible loop in \(X\). In 1949, Loewner proved that Area\((\mathbb{T}^2, g) \geq \frac{\sqrt{3}}{2}\) sys\((\mathbb{T}^2, g)^2\) for any Riemannian metric \(g\) on the 2-torus \(\mathbb{T}^2\), a result that can be viewed as a kind of reverse isoperimetric inequality.\NThe systematic study of systolic phenomena and invariants was initiated by \textit{M. Gromov} [J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037)], who proved the existence of a universal constant \(C_n\), depending only on the dimension \(n\), such that Vol\((X, g) \geq C_n\)sys\((X, g)^n\) for all \(n\)-dimensional essential polyhedra \(X\) equipped with a piecewise Riemannian metric \(g\). The present paper considers a compact connected simplicial complex of dimension 2 equipped with a piecewise Riemannian metric \(g\) and proves that if there exist classes \(\alpha, \beta\) in \(H^1(X, \mathbb{Z}_2)\) such that \(\alpha \cup \beta \not=0\) in \(H^2(X, \mathbb{Z}_2)\), then Area\((X, g) \geq \frac{1}{2}\) sys\((X, g)^2\).\N\NOne of the main motivations to study the systolic geometry of 2-dimensional complexes is its connection with finitely presentable groups. The systolic area \(\sigma (X)\) of a 2-dimensional complex \(X\) is\N\[\N\sigma (X) = \inf_g \frac{\mathrm{Area}(X, g)}{\mathrm{sys}(X, g)^2},\N\]\Nthe infimum being over all piecewise Riemannian metrics \(g\) on \(X\). Gromov introduced in [\textit{M. Gromov}, Sémin. Congr. 1, 291--362 (1996; Zbl 0877.53002)] the notion of systolic area of a finitely presentable group \(G\) as \(\sigma (G) = \inf_X\sigma (X)\), where the infimum is taken over the finite 2-dimensional simplicial complexes \(X\) with fundamental group isomorphic to \(G\).\N\textit{M. Gromov} [J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037)] showed that the systolic area of aspherical closed surfaces is at least \(\frac{3}{4}\).\N\NA group \(G\) is surface-like if there exist classes \(\alpha, \beta\) in \(H^1(G, \mathbb{Z}_2)\) such that \(\alpha \cup \beta \not=0\) in \(H^2(G, \mathbb{Z}_2)\). As the name suggests, all surface groups are surface-like by Poincaré duality. But many other groups are surface-like: free abelian groups of rank at least 2, elementary abelian 2-groups, direct and free products of a group with a surface-like group are some other examples. It is shown that for a group in this class the systolic area is bounded from below by \(\frac{1}{2}\). Also, it is established the following more general result. Let \(G\) be a group which contains a surface-like subgroup \(T\). Then, \(\sigma (G) \geq \frac{1}{2}\).