Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

Abstract

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and viceversa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions.Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length O(g 3/2 n 1/2 ) for any triangulated combinatorial surface of genus g with n triangles, and describe an O(gn)-time algorithm to compute such a decomposition.Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations. * Supported by the French ANR Blanc project ANR-12-BS02-005 (RDAM

Systolic inequalities for the number of vertices

“…In [7] and [12] it was observed that there exists a constant c ′ n such that any simplicial complex X triangulating an essential manifold has at least c ′ n sys(X) n facets (faces of dimension n). Theorem 1.4 provides a condition on a simplical complex that implies the discrete systolic inequality in terms of the vertices, (which in turn, implies the Riemannian version [7] and lower bounds for all the entries of the f -vector which in particular improve significantly the estimate on th constant c ′ n ).…”

“…In [7] and [12] it was observed that there exists a constant c ′ n such that any simplicial complex X triangulating an essential manifold has at least c ′ n sys(X) n facets (faces of dimension n). Theorem 1.4 provides a condition on a simplical complex that implies the discrete systolic inequality in terms of the vertices, (which in turn, implies the Riemannian version [7] and lower bounds for all the entries of the f -vector which in particular improve significantly the estimate on th constant c ′ n ). Unlike the estimates on the number of n-faces, lower bounds on the number of vertices (in Theorem 1.4) are not easy to derive directly from Riemannian systolic inequalities.Instead our proof adapts the approach of Larry Guth [9] and Panos Papasoglu [16], which is metaphorically referred to as "the Schoen-Yau [18] minimal hypersurface method".…”

Short cycles in high genus unicellular maps

“…the size of the smallest noncontractible cycles. The question of the systole is a well studied question in the field of geometry of surfaces (see for instance [11; 20; 21] and in particular [9] in the case of deterministic maps).…”