## Notes of Richard Schwartz’ first Cambridge lecture 03-05-2017

Pappus’s theorem and the modular group

A series of lectures with a common theme: moduli spaces of geometric objects. The geometric objects change from lecture to lecture.

1. Context

Deforming a representation ${\Gamma\rightarrow Isom(X)}$ into a larger space ${X\subset Y}$. Examples: ${Sl_2/SO(2) \subset Sl_n/SO(n)}$, ${U(1,1)/U(1)\times U(1)\subset U(2,1)/U(2)\times U(1)}$.

2. Projective geometry

Projective plane is the union of a plane and a line at infinity, can be viewed as nonzero vectors in ${{\mathbb R}^3}$ mod proportionnality. Hence a group ${PGl(3,{\mathbb R})=Gl(3,{\mathbb R})/{\mathbb R}^*}$ acting on it, simply transitively on projective frames, i.e. quadrilaterals.

The space of lines in ${{\mathbb R} P^2}$ is a dual projective space. A duality is a map fom the plane to its dual that preserves the incidence relation (i.e. composition of the duality given by a scalar product and an arbitrary projective mapping).

The flag space is the space of incident pairs of a point and a line. Topologically, it is a finite quotient of the 3-sphere.

3. Pappus’s theorem

Given 2 line, each carrying 3 points. Connect them as a complete bi-partite graph. Diagonals intersect in 3 points. Pappus state that these points belong to a line.

It is a self-dual theorem: the dual version, with points replaced with lines, is Pappus’s theorem again.

It is a special case of Pascal’s theorem on conics, which is in turn a special case of Cayley-Bacharach’ theorem on cubics.

4. The modular group

Pappus’s theorem is related to the Farey addition of fractions,

$\displaystyle \begin{array}{rcl} \frac{a}{b}\oplus\frac{c}{d}=\frac{a+c}{b+d}. \end{array}$

where the modular group enters. Indeed, start with the planar box with vertices ${(0,1/2),(0,0),(0,-1/2),(1,1/2),(1,0),(1,-1/2)}$. Apply Pappus to vertical sides, get vertical segment at abcissa ${1/2}$, which is the Farey sum of ${0}$ and ${1}$. Iterate on both sides, get successively all rational abcissae in Farey order.

Each Farey addition corresponds to a directed edge of the Farey graph (sides of the tesselation of the disk by ideal triangles). The symmetry group of the tiling is generate by ${i}$ (switching orientation), ${t}$ and ${b}$ (horospherical translations), subject to ${i^2=1}$, ${tit=b}$, ${bib=t}$, ${tibi=biti=1}$

5. Marked boxes

Next I describe the operations related to Pappus’s theorem corresponding to generators ${i,t,b}$ of modular group.

A box is a convex quadrilateral with two marked points on two opposite side, one called bottom and the other top (this makes sense over any ordered fields). Applying Pappus splits the initial box into two boxes, called ${b}$ and ${t}$. ${i}$ switches the two opposite sides.

On the standard box defined above, the operation has the same effect as the modular group acting on rationals.

A marked box depends on 2 parameters, specifying the positions of marked points on respective sides. These parameters are unchanged under ${i}$, ${b}$ and ${t}$.

Lemma 1 For all marked boxes ${X}$ there exists an order 3 projective transformation ${T}$ mapping ${t(X)}$ to ${b(X)}$ to ${i(X)}$ to ${t(X)}$ cyclically.

Next I describe the corresponding element of order 2 of the modular group (viewed as a free product of ${{\mathbb Z}/3{\mathbb Z}}$ and ${{\mathbb Z}/2{\mathbb Z}}$).

Lemma 2 For all marked boxes ${X}$ there exists a duality mapping ${X}$ to ${i(X)}$ to ${X}$ cyclically.

6. Representations

Changing box parameters amounts to perturbing the initial representation of the modular group in the group of automorphisms of the flag variety (group of dualities), which is a semi-direct product of ${{\mathbb Z}/2{\mathbb Z}}$ with ${PGl(3,{\mathbb R})}$.

Barbot-Lee-Valerio (2016) show that these representations are limits of Anosov representations of pairs of pants groups. Kapovich shows that these representations are relatively Anosov.

I show that these representations are discrete, they have a domain of discontinuity, whose quotient is the trefoil knot complement.

Fix a vertex of the Farey tesselation. Join it to rational points of the projective line. To each such segment, there corresponds a flag, obtained by iterating Pappus box division. This map from a dense subset of the projective line to a subset of the flag manifold extends by continuity. Indeed, this amounts to showing that certain diamonds (quadrilaterals) get small.

One gets a fractal curve in the flag manifold, carrying a fractal line field. When projected to projective plane, each line of the line field intersects the curve at a single point (this is the only line field with this property). Experiments indicate that the box dimension can be as large as 1.3.

The domain of discontinuity is the set of pairs ${(p,\ell)}$ where ${p}$ does not belong to the curve and ${\ell}$ does not belong to the line field. The topology does not depend on parameters ${s}$ and ${t}$.

7. Questions

Curt McMullen has been able to compute the dimension of certain self-affine carpets. Maybe one can compute the dimension here. One cannot expect a connection with a Poincare series.

Probably, there are more parabolic preserving representations of the modular group in ${PGl(3,{\mathbb R})}$.