ThomasBudzinski(UBC) …budzinski/slides/talk... · 2019-10-25 ·...

On the minimal diameter of hyperbolic surfaces

Thomas Budzinski (UBC)Joint work with Nicolas Curien (Université Paris–Saclay) and

Bram Petri (Sorbonne Université)

October 23rd 2019UBC Probability Seminar

Thomas Budzinski Minimal diameter of hyperbolic surfaces

Introduction

Goal: study the minimal possible diameter of hyperbolicsurfaces with high genus g : asymptotic to 1× log g .

Small diameter ≈ highly connected objects.Random (say, 3-regular) graphs are also very connected:probabilistic method, very common in combinatorics.Probabilistic method in hyperbolic geometry.


Program

I The diameter of 3-regular random grapsII Notions of hyperbolic geometryIII The diameter of hyperbolic surfacesIV Ideas of the proofV Perspectives


The diameter of 3-regular random graphs

Gn obtained from n vertices with 3 half-edges each, bymatching the half-edges uniformly at random (connected withproba 1− O(1/n)).Diameter: maximal graph distance between two vertices.

Theorem (Bollobas–Fernandez de la Vega, 1982)

diam(Gn)

log2 n

(P)−−−−→n→+∞

1.

Lower bound: a ball of radius r has size at most 3× 2r , so thediameter is ≥ log2 n.


Diameter of random graphs: proof

Upper bound: it is enough to prove that for any two fixedvertices v1, v2:

P ((1+ ε) log2 n ≤ dGn(v1, v2) < +∞) = o

(1n2

).

Explore balls of radius r = 1+ε2 log2 n around v1 and v2, and

try to connect them.

v1 ?

If no "bad step", we would have |∂Br (v1)| = 3× 2r = 3n1+ε2 .



But P (bad step at time i) ≤ i+23n , with independence over i .

Consequence: with probability 1− o( 1n2

):

O(1) bad steps in the ball of radius 1−ε2 log2 n bad steps,

o(n

1+ε2

)bad steps between distances 1−ε

2 log2 n and1+ε2 log2 n,

so |∂Br (v1)| ≥ δn1+ε2 w.h.p., and the same is true around v2.

Br (v1) Br (v2)



If Br (v1) ∩ Br (v2) 6= ∅, we are done.If not, each loose half-edge on ∂Br (v1) has probability|∂Br (v2)|

n ≥ δn−1−ε2 to be connected to Br (v2). So

P (Br (v1) and Br (v2) not directly linked) ≤(1− δn−

1−ε2

)n 1+ε2

≤ exp (−δnε)

= o

(1n2

),

and dGn(v1, v2) ≤ 2r + 1 ≤ (1+ ε) log2 n with very highprobability.


Hyperbolic geometry

The hyperbolic plane H can be seen as the unit disk, equippedwith the metric

ds2 =4dx2

1− |x |2.

Curvature: |Bε(x)| = πε2 − π12ε

4K (x) + o(ε4).Riemann uniformization theorem: H is the unique simplyconnected surface with constant curvature equal to −1.


Compact hyperbolic surfaces

A compact hyperbolic surface S is a 2d manifold equippedwith a Riemannian metric with constant curvature −1. Weconsider closed surfaces, i.e. no boundary.Gauss–Bonnet formula:

∫S K (x)dx = 2π(2− 2g), where g is

the genus of the surface, i.e. the number of holes. So g ≥ 2.

S

Equivalent definitions:S is locally isometric to H,S is a quotient of H (by a nice enough group action),S is a surface equipped with a conformal structure.


Pairs of pants

Existence, but no uniqueness: for g ≥ 2, hyperbolic metrics ona genus g surface form a (6g − 6)-dimensional space Mg

called the moduli space.One way to build a lot of them is to use pants.For any `1, `2, `3 ≥ 0, there is a unique surface isomorphic tothe sphere minus 3 disjoint disks, such that:

the boundaries of the three disks are closed geodesics withlengths `1, `2, `3;the curvature is −1 outside of the boundary.

`1

`2`3


Gluings of pants

By gluing 2g − 2 pairs of pants such that the lengths of theboundaries match two by two, we can build many hyperbolicsurfaces.6g − 6 degrees of freedom: 3g − 3 for the lengths of thecycles, and 3g − 3 for the twists.Conversely, every hyperbolic surface of genus g can be cut by3g − 3 closed geodesics into 2g − 2 pairs of pants.


Interesting quantities for hyperbolic surfaces

Given a hyperbolic surface S , several natural quantities to lookat, and try to optimize over the moduli space:

diameter,spectral gap (eigenvalues of the Laplacian),Cheeger constant (isoperimetric inequalities),systole (length of the smallest closed geodesics).

All of these measure the "connectivity" of the surface.In the context of hyperbolic surfaces, non-optimal bounds(constant factors) often obtained via arithmetic constructions[Brooks, Buser, Kim, Sarnak...].A typical graph is very connected, so random graphs (likeuniform 3-regular graphs) are close to optimal for thesequantities.


Diameter of hyperbolic surfaces

Diameter: maximal distance between two points of S .Easy: supS∈Mg

diam(S) = +∞.

×2 −→

ε

ε ε

log 1ε

Lower bound for the minimal diameter: volume growthargument

By Gauss-Bonnet, Area(S) = 2π(2g − 2).In H, the area of balls is Area(Br (x)) = 2π(cosh(r)− 1).S is a quotient of H, so Area(Br (x)) ≤ 2π(cosh(r)− 1) in S .So if Br (x) covers S , then cosh(r)− 1 ≥ 2g − 2, so

infS∈Mg

diam(S) ≥ cosh−1(2g − 1) = log g + O(1).

Best lower bound [Bavard 1996]: also log g + O(1).


Main theorem

Theorem (B.–Curien–Petri, 2019)

We haveminS∈Mg

diam(S) = (1+ o(1)) log g .

Construction: random gluing of pants!Start from 2g − 2 pants with perimeters (a, a, a), and glue the6g − 6 holes uniformly at random to obtain Sg ,a.Twist 0: the "centers" of two neighbour pants have the sameprojections on the boundary.

We show diam (Sg ,a) ∼ 1βa

log g w.h.p., where βa < 1, butβa → 1 as a→∞.


A crude bound

For a fixed, the diameter of a pair of pants is a constant.Enough to bound distances between the centers of the pants.Quick bound: diam (Sa,g ) ≤ 2dadiam (G2g−2) ∼

n→∞2da log2 n.

After computing da for a→ +∞, we get ≈ 1.38 log g .

a

a a

c da

c1

c2

c3

c4

Not optimal: sometimes, there is a much shorter path.


Adapting the explorations

Instead of using random graphs as a black box, adapt theproof!Adapt the exploration to the hyperbolic metric, instead of thegraph distance.Ideal situation: the neighbourhood of one center looks like aninfinite tree of pants Ta. We need to understand its growth!


Growth of the infinite tree of pants

Br : ball of radius r around a center of the pants tree Ta. Let|Br | be the number of pants whose center is in Br .

Lemma

We have |Br (Ta)| ∼r→+∞

Caeβar , where βa → 1 as a→ +∞.

Sketch of proof: pants can be decomposed in two right-angledhexagons.

−→a a

a

bb

b

a/2 a/2

a/2

bb

b

Gluings with twist 0, so the red "weldings" match onneighbour pants.


Growth of the infinite pair of pants

Hence, the tree of pants is the gluing of two copies of aninfinite tree of right-angled hyperbolic hexagons:

Above: infinite tree of hexagons for increasing values of a.The growth of hexagon trees corresponds to orbital countingfor a subgroup of PSL2(R) generated by reflexions. This is wellunderstood by geometers [Patterson–Sullivan, McMullen...]


Bounding the diameter

As for graphs, we want to show, for any centers c1, c2:

P(dhyp(c1, c2) ≥

(1+ ε

βa

)log g

)= o

(1g2

).

We explore the balls of radius r = 1+ε2βa log g around c1 and c2

for the hyperbolic metric on the infinite tree of pants.As for graphs, we can bound the number of "bad" steps: thevolume and boundary of Br (c1) are at least a constant timeswhat they would be in the tree of pants.

So |∂Br (c1)| ≥ δ exp(βa

1+ε2βa log g

)= δg

1+ε2 , and the same is

true for c2.As for graphs, this implies that with very high probability,there is an edge between Br (c1) and Br (c2), sodhyp(c1, c2) ≤ 2r + O(1) = 1+ε

βalog g + O(1).


Perspectives

Error term? For random graphs O (log log n). It should also betrue here (for a = log log n).Other natural models of random surfaces:

Brooks–Makover surfaces (built by uniformizing randomtriangulations with unconstrained genus): diameter ∼ 2 log g[BCP 2019+],Weil–Petersson random surfaces (≈ "Lebesgue measure" onthe space of hyperbolic surfaces): diameter ≤ 40 log g[Mirzakhani 2013].

Study other quantities? Maximal spectral gap for hyperbolicsurfaces? Cheeger constant?Hyperbolic manifolds in higher dimensions?


THANK YOU !


ThomasBudzinski(UBC) …budzinski/slides/talk... · 2019-10-25 ·...

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