arXiv research

A new systolic inequality for mod 2 systoles is established.

A new method using mod n covering improves systolic inequalities.

We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".

We adapt the theory of currents in metric spaces, as developed by the first-mentioned author in collaboration with B. Kirchheim, to currents with coefficients in Z_p. Building on S. Wenger's work in the orientable case, we obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these inequalities to prov…

Let $\mathrm{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. Given a finite subgroup $H$ of $\mathrm{Mod}(S_g)$, let $\mathrm{Fix}(H)$ denote the set of fixed points induced by the action of $H$ on the Teichmüller space $\mathrm{Teich}(S_g)$. When $H$ is cyclic with …

A recent preprint of S. Kojima and G. McShane [KM] observes a beautiful explicit connection between Teichmüller translation distance and hyperbolic volume. It relies on a key estimate which we supply here: using geometric inflexibility of hyperbolic 3-manifolds, we show that for $S$ a closed surface, and $ψ\in \text{Mo…

The paper proves optimal systolic inequalities for Möbius strip and Klein bottle.

Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.

A new systolic inequality with a remainder for the real projective plane.

We prove a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole and the codimension 1 systole with coefficients in Z_2. As an application, we prove that Lusternik-Schnirelmann category and systolic category agree for non-orientable closed manifolds of dimension 3, extending our…

In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the systolic volume depen…

In this paper we investigate the systolic landscape of translation surfaces for fixed genus and fixed angles of their cone points. We furthermore study how the systoles of a translation surface relate to the systoles of its graph of saddle connections. This allows us to develop an algorithm to compute the systolic rati…

Max systoles on spheres with punctures are counted.

Study on complexity of systolic geodesics on Bolza surface.

New finding links hyperbolic manifold systolic volume to triangulation complexity.

Upper bound found for systolic geometry on manifolds with positive scalar curvature.

New manifolds with small systoles not quasi-arithmetic.

Sharp inequalities for Kähler manifolds' systolic invariants are established.

Systoles of hyperbolic manifolds are dense and related to Salem numbers.

Random surfaces with long systoles created from graph theory ideas.

Upper bounds found for systole function critical points on surface moduli space.

The study of systoles in arithmetic hyperbolic manifolds.

Extremal length systole is maximized at the Bolza surface.

For all systolic groups we construct boundaries which are EZ--structures. This implies the Novikov conjecture for torsion--free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex, which we prove to have coarsely similar properties to geodesics in CAT(0) spaces.

New systolic inequality for 3D contact forms on Seifert bundles.

The aim of this text is to present the concept of systole of a compact riemannian manifold and to give an overview of systolic geometry. I will also present the "regularization technique", which leads to major results in systolic geometry. I will detail how this technique allows to link the systolic volume of some clos…

The paper analyzes systoles of complex projective spaces under various metrics.

New Finsler metric on sphere disproves systolic ratio conjecture.

The study finds a unique systole maximum in non-hyperelliptic surfaces.

Extends Gromov's optimal systolic inequality to manifolds with specific cohomology properties.

Study finds bounds for systole length on arithmetic punctured spheres.

The study finds large systoles in translation surfaces and hyperelliptic ones.

Sharp systolic inequality for invariant tight contact forms on S1-bundles over S2.

Directly proves logarithmic systolic growth for all hyperbolic surfaces.

Study shows systole behavior changes significantly for large genus hyperbolic surfaces.

New links in 3-manifolds have large systole.

Paper proves optimal systolic inequality for manifolds with positive triRic curvature.

We give the first example of systolic freedom over torsion coefficients. The phenomenon is a bit unexpected (contrary to a conjecture of Gromov's) and more delicate than systolic freedom over the integers.

Proves upper bound on systolic ratio for circle fillings.

Study systoles and diameters on hyperbolic surfaces, finding an upper bound for their ratio.

We prove optimal systolic inequalities on Finsler Mobius bands relating the systole and the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimal systolic in- equality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric…

New index characterizes non-smooth Zoll convex bodies.

For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation be…

We show that for closed orientable manifolds the $k$-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree $k$ that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles…

We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.

We show that the systolic constant, the minimal entropy, and the spherical volume of a manifold depend only on the image of the fundamental class under the classifying map of the universal covering. Moreover, we compute the systolic constant of manifolds with fundamental group of order two (modulo the value on the real…

We prove a finiteness result for the systolic area of groups, answering a question of M. Gromov. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of~2-complexes whose systolic area is uniformly bounded. Furthermore, we prove a uniform systolic inequality for all 2-complexe…

Study on systole of random hyperbolic 3-manifolds, proving limit exists and calculating it.