## Notes of Richard Schwartz’ third Cambridge lecture 17-05-2017

Iterated barycentric subdivisions and steerable semi-groups

In two dimensions, there are many different affinely natural procedures on simplices: the barycentric subdivision (defined by coning and induction on dimension), yielding 6 triangles; the truncation of corners, yielding 4 triangles. The second is better behaved for numerics (triangles do not get thin, they stay similar to each other, in fact only 2 shapes are encountered).

Question. What shapes of simplices arise in an iterated subdivision scheme?

Diaconis and McMullen show that almost every triangle produced gets thin.

1. The space of shapes

I am aiming at a different question. Shapes form a topological space. Let ${X^n}$ be the space of labelled ${n}$-simplices mod scaling. Say we normalize so that volume stays equal to 1. Then ${X^n}$ is a principal homogeneous space of ${Sl(n,{\mathbb R})}$. An affinely natural subdivision process amounts to a subdivision of the standard simplex.

Question. Does the iteration scheme produce a dense set of shapes?

The answer is positive in 2 dimensions (Barany-Beardon-Carne 1990). I gave a positive answer in 3 and 4 dimensions, I will explain my solution. My guess is that answer should be negative for ${n}$ large enough.

The question is equivalent to the density of the semi-group generated by the matrices that map the standard simplex to each of the labelled tiles.

Lemma: the subgroup ${S}$ generated by the subdivision is either discrete or dense.

The proof is not very enlightening. It follows that the subgroup ${\langle S\rangle}$ generated by the subdivision is dense.

Lemma: If ${S}$ contains a bounded infinite walk, then the semi-group ${S}$ is dense in ${\langle S\rangle}$. Indeed, a long bounded walk contains segments ${g_1\cdots g_k}$ which are close to 1, hence an expression of inverses as elements of the semi-group ${S: g_1^{-1}=g_2 \dots g_n}$ up to small error.

1.1. ${n=2}$

If you label right, then all generators are infinite order elliptics.

1.2. ${n=3}$

A computer search revals some infinite order elliptic elements. They represent a positive fraction of words of length 3.

Experiments suggest existence of an infinite walk up to ${n=10}$.

2. Proof

Look at horoballs in ${X}$. Fix an origin ${o\in X}$.

Definition 1 A horoball is special if it contains the origin ${o}$ in its boundary.

A set ${W\subset X}$ is a strong wheel if every special horoball contains a point of ${W}$ in its interior.

A semi-group ${S}$ is steerable if the orbit ${So}$ contains a steering wheel.

Lemma: If ${S}$ is steerable, then ${S}$ has a bounded infinite walk.

The idea is that if a walk goes far away from ${o}$, use a special horoball containing it in its boundary, pick a point there, start again.

Therefore, the point is to exhibit a steering wheel.

Lemma: Consider the Hadamard map ${(X,o)\rightarrow H^N}$, ${N=\frac{n(n+1)}{2}-1}$ (geodesic polar coordinates) preserving min sectional curvature. This is distance non-decreasing. This allows to carry the problem to 9-dimensional hyperbolic space.

The wheel is a finite set of 144 points on the boundary of some ball ${B}$ centered at ${o}$. To certify that they form a steering wheel, apply the following criterion: each of them points towards the center of a special horoball. If the convex hull of these centers contains ${B}$, then the horoballs cover ${\partial B}$, hence the wheel.

3. Questions

What makes you think the answer should be negative in high dimensions? Because ${n!}$ is too small a number of horospheres to cover a sphere of dimension ${N=\frac{n(n+1)}{2}-1}$. Experiments indicate that this phenomenon could start at ${n=9}$.

The semi-group would be uniformly discrete?