scholarly journals Graphs of systoles on hyperbolic surfaces

2019 ◽  
Vol 11 (01) ◽  
pp. 1-20 ◽  
Author(s):  
Bidyut Sanki ◽  
Siddhartha Gadgil
Keyword(s):  
Planar Graphs ◽  
Classical Result ◽  
Hyperbolic Surface ◽  
Necessary Condition ◽  
Major Result ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Fat Graphs ◽  
Admissible Graph

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible).There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient.It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

scholarly journals Short closed geodesics with self-intersections

2020 ◽  
Vol 169 (3) ◽  
pp. 623-638
Author(s):  
VIVEKA ERLANDSSON ◽  
HUGO PARLIER
Keyword(s):  
Linear Function ◽  
Hyperbolic Surface ◽  
Minimal Length ◽  
The Self ◽  
Closed Geodesics ◽  
Intersection Numbers ◽  
Hyperbolic Surfaces ◽  
Fixed Integer

AbstractOur main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

SYSTOLIC FILLINGS OF SURFACES

2018 ◽  
Vol 98 (3) ◽  
pp. 502-511
Author(s):  
BIDYUT SANKI
Keyword(s):  
Closed Geodesic ◽  
Disjoint Union ◽  
Hyperbolic Surface ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Simple Closed Geodesics

A filling of a closed hyperbolic surface is a set of simple closed geodesics whose complement is a disjoint union of hyperbolic polygons. The systolic length is the length of a shortest essential closed geodesic on the surface. A geodesic is called systolic, if the systolic length is realised by its length. For every $g\geq 2$, we construct closed hyperbolic surfaces of genus $g$ whose systolic geodesics fill the surfaces with complements consisting of only two components. Finally, we remark that one can deform the surfaces obtained to increase the systole.

scholarly journals EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES

2019 ◽  
Vol 99 (03) ◽  
pp. 508-520
Author(s):  
BIDYUT SANKI
Keyword(s):  
Euler Characteristic ◽  
Isometric Embedding ◽  
Hyperbolic Surface ◽  
Hyperbolic Surfaces ◽  
Metric Graphs ◽  
Metric Graph ◽  
Complementary Region

An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$ , there is an embedding of $(G,d)$ (possibly after a rescaling of $d$ ) on a surface of genus $g$ . Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$ . The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$ , where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.

scholarly journals On bounding the Stückrad-Vogel multiplicity

1991 ◽  
Vol 34 (2) ◽  
pp. 251-257 ◽  
Author(s):  
L. O'Carroll
Keyword(s):  
Classical Result ◽  
Tensor Products ◽  
Point Of View ◽  
Necessary Condition

Using a classical result of Nagata, Achilles, Huneke and Vogel gave a criterion for the Stückrad-Vogel multiplicity to take the value one. We use Huneke's extension of Nagata's theorem to give a necessary condition for the Stückrad-Vogel multiplicity to have an arbitrary preassigned bound, under certain conditions. A usable criterion of multiplicity n results (given mild hypotheses). We also revisit some basic results in the Stückrad-Vogel theory in the light of the behaviour of tensor products of affine primary rings, and also revisit some arguments of Achilles, Huneke and Vogel from the point of view of fibre rings.

scholarly journals Geodesics in Margulis spacetimes

2011 ◽  
Vol 32 (2) ◽  
pp. 643-651 ◽  
Author(s):  
WILLIAM M. GOLDMAN ◽  
FRANÇOIS LABOURIE
Keyword(s):  
Compact Convex ◽  
Hyperbolic Surface ◽  
Orbit Equivalence ◽  
Closed Geodesics ◽  
Timelike Geodesic ◽  
Convex Core

AbstractLet M3 be a Margulis spacetime whose associated complete hyperbolic surface Σ2 has a compact convex core. Generalizing the correspondence between closed geodesics on M3 and closed geodesics on Σ2, we establish an orbit equivalence between recurrent spacelike geodesics on M3 and recurrent geodesics on Σ2. In contrast, no timelike geodesic recurs in either forward or backward time.

scholarly journals The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces.With the appendix by Khalid Bou–Rabee

2017 ◽  
Vol 166 (1) ◽  
pp. 83-121
Author(s):  
NEHA GUPTA ◽  
ILYA KAPOVICH
Keyword(s):  
Lower Bounds ◽  
Free Group ◽  
Growth Function ◽  
Residual Finiteness ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Finitely Generated ◽  
Closed Curves ◽  
Index Function ◽  
Finite Covers

AbstractMotivated by the results of Scott and Patel about “untangling” closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou–Rabee connects the primitivity index functionfprim(n,FN) to the residual finiteness growth function forFN.

scholarly journals Density of half-horocycles on geometrically infinite hyperbolic surfaces

2012 ◽  
Vol 33 (4) ◽  
pp. 1162-1177
Author(s):  
BARBARA SCHAPIRA
Keyword(s):  
Tangent Bundle ◽  
Hyperbolic Surface ◽  
Hyperbolic Surfaces ◽  
Weak Assumption ◽  
Unit Tangent Bundle

AbstractOn the unit tangent bundle of a hyperbolic surface, we study the density of positive orbits $(h^s v)_{s\ge 0}$ under the horocyclic flow. More precisely, given a full orbit $(h^sv)_{s\in {\mathbb R}}$, we prove that under a weak assumption on the vector $v$, both half-orbits $(h^sv)_{s\ge 0}$ and $(h^s v)_{s\le 0}$ are simultaneously dense or not in the non-wandering set $\mathcal {E}$of the horocyclic flow. We give also a counterexample to this result when this assumption is not satisfied.

scholarly journals VC-dimensions of random function classes

2008 ◽  
Vol Vol. 10 no. 1 (Combinatorics) ◽  
Author(s):  
Bernard Ycart ◽  
Joel Ratsaby
Keyword(s):  
Random Function ◽  
Classical Result ◽  
High Probability ◽  
Necessary Condition ◽  
Vc Dimension ◽  
Uniform Sampling ◽  
Function Classes ◽  
International Audience ◽  
Asymptotic Probability ◽  
Binary Functions

Combinatorics International audience For any class of binary functions on [n]={1, ..., n} a classical result by Sauer states a sufficient condition for its VC-dimension to be at least d: its cardinality should be at least O(nd-1). A necessary condition is that its cardinality be at least 2d (which is O(1) with respect to n). How does the size of a 'typical' class of VC-dimension d compare to these two extreme thresholds ? To answer this, we consider classes generated randomly by two methods, repeated biased coin flips on the n-dimensional hypercube or uniform sampling over the space of all possible classes of cardinality k on [n]. As it turns out, the typical behavior of such classes is much more similar to the necessary condition; the cardinality k need only be larger than a threshold of 2d for its VC-dimension to be at least d with high probability. If its expected size is greater than a threshold of O(&log;n) (which is still significantly smaller than the sufficient size of O(nd-1)) then it shatters every set of size d with high probability. The behavior in the neighborhood of these thresholds is described by the asymptotic probability distribution of the VC-dimension and of the largest d such that all sets of size d are shattered.

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