**Rigidity and Buildings**

**Dave Constantine**

**Geometry and Dynamics**

I study geometry and dynamics, in particular, ways to use dynamics to answer geometric questions. Some of these questions fall under the general rubric of ‘rigidity.’

**What is Rigidity?**

Theorem (Mostow Rigidity Theorem, 1968):

Among

compact hyperbolic manifolds of dimension at least 3

,

Floppy, topological data ==> complete, rigid, geometric description.

**Quasi-isometric Rigidity**

What do we mean when we say two geometric objects X and Y are ‘the same?’

**dconstantine@wesleyan.edu**

**NSM lunch seminar**

**Nov 8, 2013**

Plan for Today:

Introduce a new type of geometric object -- a building.

Discuss two types of rigidity, including some recent work.

Convince you to care?

or

**What is a building?**

a case of mathematical terminology run wild

Buildings have

apartments

,

chambers

,

panels

and

walls

.

Each apartment is a copy of Euclidean space tiled by polygonal chambers. (For a Euclidean building.)

There are many apartments, which overlap and branch apart from one another along the walls:

The apartments of a Euclidean building are copies of tiled Euclidean space.

In this talk, we’ll also be interested in Fuchsian buildings, whose apartments are copies of tiled hyperbolic space:

(B1)

Each apartment is a copy of tiled space (Euclidean, hyperbolic, or spherical).

(B2)

Any two chambers belong to a common apartment.

(B3)

Whenever two apartments overlap, one can be ‘folded’ onto the other, matching them up exactly and not disturbing their overlap.

Often we add:

(B4)

Branching occurs at every wall.

These complicated objects actually obey three simple axioms:

(B1)

Each apartment is a copy of tiled space:

Each apartment is regular and orderly.

(B2) Any two chambers belong to a common apartment:

Why would you need privacy, comrade?

(B3) Overlapping apartments can be ‘folded’ one onto the other:

All ‘property’ is equal and interchangeable.

Think of a building as a Soviet-era apartment block:

Why Buildings?

Buildings have lots of symmetry. (See the ‘folding’ guaranteed by Axiom (B3).)

This symmetry is naturally related to some important algebraic structures.

(Lie groups over the p-adics, parabolic subgroups of semisimple Lie groups...)

They generalize (in an interesting way) Euclidean and hyperbolic space.

There is an interplay of the continuous and the discrete in their geometry.

(Euclidean and Fuchsian)

buildings are infinite – they extend forever.

Sometimes, one can use the symmetries of the building to ‘roll it up’ into an object of finite size.

We will call these finite objects compact building quotients.

A note for later:

Non-rigidity:

Rigidity theorems have

this general structure:

Theorem:

Among [some class of (usually) geometric objects],

[some seemingly insufficient data] ==>

[a full description of the object].

A rigidity theorem is a classification theorem, plus an element of surprise, plus some genetic resemblance to Mostow’s theorem and the many results it has inspired.

Definition:

X and Y are

isometric

if there is a function f : X → Y

matching them up point-for-point such that

d(f(x1), f(x2)) = d(x1, x2)

for any two points x1, x2 in X.

I.e. all distances are exactly the same, so

the two objects look exactly alike.

Isometry is a very strong condition:

Mathematically, it can be hard to prove or verify.

Physically, it’s unreasonable that all distances match exactly.

1/K d(x1,x2) − C ≤ d(f(x1),f(x2)) ≤ K d(x1,x2) + C

Definition:

A

quasi-isometry

is a map f such that

1/K d(x1,x2) − C ≤ d(f(x1),f(x2)) ≤ K d(x1,x2) + C

for any pair of points, and for some constants K > 1, C > 0 (which are independent of the points chosen).

(Plus some sort of ‘coarsely bijective’ assumption.)

Think of a quasi-isometry in terms of vision:

The factor C means you cannot determine exactly where points are up to an error of size C.

Your vision is blurry.

The factor K means that distances can be stretched or contracted by a factor of K.

You have astigmatism.

The quasi-isometric rigidity question:

If two spaces in some class are quasi-isometric, are they actually isometric?

or, in the terms of our metaphor,

Can you, with your blurry, astigmatic vision (no matter how blurry or astigmatic we make it) tell different geometric objects in this class apart?

Example:

After blurring and smudging (and changing colors), which one is Ishita?

=

=

Conclusion: Ishita is not quasi-isometrically rigid (among the class of Wesleyan employees).

Quasi-isometric rigidity for buildings

Theorem (Kleiner-Leeb):

(Thick, irreducible, Moufang boundary)

Euclidean buildings are quasi-isometrically rigid (up to scaling).

Proof:

Look at the building from a long way off. The blurriness in your vision makes no difference at large scales. Despite your astigmatism, it is still possible to detect the branching structure of the building and distinguish different buildings.

Fuchsian buildings are not quasi-isometrically rigid, but

Theorem (Xie):

Quasi-isometric Fuchsian buildings are

combinatorially isomorphic

.

Open question: Can the assumption of ‘Moufang boundary’ be removed from [Kleiner-Leeb]?

**Marked length spectrum**

The marked length spectrum records a specific set of distances in our geometric object:

Marked length spectrum rigidity

Theorem (Otal, Croke; Hamenstadt)

If two (

homeomorphic

) objects in one of the classes below have the same marked length spectrum, then they are isometric:

Compact, non-positively curved surfaces.

(e.g. the picture on the previous slide)

Compact, negatively curved locally symmetric spaces.

Definition:

The

marked length spectrum

records of the length of each minimal-length loop (or each closed geodesic) in the space

(and associates this length to its free homotopy class)

.

A major open question is whether MLS rigidity holds for compact, non-positively curved manifolds of dimension > 2.

Recently, Jean-Francois Lafont

(Ohio State) and I have proven:

Theorem:

MLS rigidity holds for

Compact quotients of Fuchsian buildings.

Compact (

geodesic metric

) spaces of (

topological

) dimension 1.

Proof for Fuchsian buildings:

One argument is modeled on Otal and Croke’s

approach, and uses the geodesic flow (←dynamics!), as well as Xie’s result.

Proof for 1-dimensional spaces:

2L(

green

) = L(

green

and

blue

) + L(

green

and

red

)

− L(

blue

and

red

)

Ex: MLS rigidity for ellipsoids

You are given the lengths of all closed loops on an ellipsoid. Can you determine its geometry?

Exercise: MLS rigidity for tori.