Posted by: matheuscmss | October 29, 2016

## Lyapunov exponents of regular origamis are not determined by commutators

Finite groups generated by two elements are a rich source of examples of origamis (square-tiled surfaces). Indeed, given a finite group ${G}$ generated by ${h}$ and ${v}$, we take a collection of unit squares ${Sq(g)\subset\mathbb{R}^2}$ indexed by the elements ${g\in G}$ and we glue by translations the rightmost vertical, resp. topmost horizontal, side of ${Sq(g)}$ with the leftmost vertical, resp. bottommost horizontal, side of ${Sq(gh)}$, resp. ${Sq(gv)}$, to obtain an origami ${M(G,h,v)}$ naturally associated to the data of ${(G, h, v)}$.

Such origamis were baptized regular origamis by David Zmiaikou in his PhD thesis and the study of the so-called Kontsevich-Zorich over regular (and quasi-regular) origamis was pursued in our joint article with Jean-Christophe Yoccoz.

In this short post, I would like discuss some informal questions posed by Jean-Christophe right after the completion of our joint paper with David Zmiaikou (in order to keep a trace of them in case someone [e.g., myself] wants to think about this matter in the future).

1. Regular origamis and commutators

The commutator ${c=[h,v]=hvh^{-1}v^{-1}}$ determines the nature of the conical singularities of the origami ${M(G,h,v)}$: in fact, ${M(G, h, v)}$ has exactly ${\#G/\textrm{order}(c)}$ such singularities and the total angle around each of them singularities is ${2\pi \cdot \textrm{order}(c)}$.

Also, Yoccoz, Zmiaikou and I showed that the “potential blocks” of the Kontsevich-Zorich cocycle over the ${SL(2,\mathbb{R})}$-orbit of ${M(G,h,v)}$ are completely determined by the commutator ${c=[h,v]}$ (cf. Theorem 1.1 of our paper).

Given this scenario, Jean-Christophe asked whether the Lyapunov exponents of the Kontsevich-Zorich cocycle of ${M(G,h,v)}$ were also completely determined from the knowledge of ${c=[h,v]}$.

2. Lyapunov exponents and commutators

By the time we finished our joint paper, David Zmiaikou pointed out to me that the commutator is not a complete invariant for many algebraic question about ${(G,h,v)}$: for example, Daniel Stork proved (among other things) that the pairs of permutations ${\sigma:=(136), \tau:=(12345)}$ and ${\sigma'=(13)(26), \tau'=(12645)}$ have the same commutator ${c=(13642)=[\sigma,\tau]=[\sigma',\tau']}$ but they generate distinct T-systems of the alternate group ${A_6}$ (cf. Section 2.6 of David Zmiaikou’s PhD thesis for more comments).

This remark led me to play with the origamis ${\mathcal{O}:=M(A_6,\sigma,\tau)}$ and ${\mathcal{O}':=M(A_6,\sigma',\tau')}$ in an attempt to answer Jean-Christophe’s question.

First, note that both of them have ${\#A_6/\textrm{order}(c) = 72}$ conical singularities and the total angle around each of them is ${2\pi\cdot \textrm{order}(c)=10\pi}$. In particular, both ${\mathcal{O}}$ and ${\mathcal{O}'}$ have genus ${145=(\frac{72\times 4}{2}+1)}$.

On the other hand, a short computation with the aid of Sage (or a long calculation by hand) reveals that the ${SL(2,\mathbb{Z})}$-orbit of ${\mathcal{O}}$ has cardinality ${15}$ and the ${SL(2,\mathbb{Z})}$-orbit of ${\mathcal{O}'}$ has cardinality ${10}$ (this is closely related to the comments in Section 5 of Stork’s paper).

Remark 1 Recall that it is easy to algorithmically compute ${SL(2,\mathbb{Z})}$-orbits of origamis described by two permutations ${h}$ and ${v}$ of a finite collection of squares because ${SL(2,\mathbb{Z})}$ is generated by ${A=\left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right)}$ and ${B=\left(\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array}\right)}$, and these matrices act on pairs of permutations by ${A(h,v)=(h,vh^{-1})}$ and ${B(h,v)=(hv^{-1},v)}$ (and the permutations ${(h,v)}$ and ${(shs^{-1},svs^{-1})}$ generate the same origami).

Moreover, this calculation also reveals that

• the ${SL(2,\mathbb{R})}$-orbit of ${\mathcal{O}}$ decomposes into four ${A}$-orbits:
• two ${A}$-orbits have size ${3}$ and all origamis in these orbits decompose into ${120}$ horizontal cylinders of width ${3}$ and height ${1}$;
• one ${A}$-orbit has size ${4}$ and all origamis in this orbit decompose into ${90}$ horizontal cylinders of width ${4}$ and height ${1}$;
• one ${A}$-orbit has size ${5}$ and all origamis in this orbit decompose into ${72}$ horizontal cylinders of width ${5}$ and height ${1}$.
• the ${SL(2,\mathbb{R})}$-orbit of ${\mathcal{O}'}$ decomposes into three ${A}$-orbits:
• one ${A}$-orbit contains a single origami decomposing into ${180}$ horizontal cylinders of width ${2}$ and height ${1}$;
• one ${A}$-orbit has size ${4}$ and all origamis in this orbit decompose into ${90}$ horizontal cylinders of width ${4}$ and height ${1}$;
• one ${A}$-orbit has size ${5}$ and all origamis in this orbit decompose into ${72}$ horizontal cylinders of width ${5}$ and height ${1}$.

This information can be plugged into the Eskin-Kontsevich-Zorich formula to determine the sums ${L(\mathcal{O})}$ and ${L(\mathcal{O}')}$ of the non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle over the ${SL(2,\mathbb{R})}$-orbits of ${\mathcal{O}}$ and ${\mathcal{O}'}$.

Indeed, if ${M}$ is an origami with ${\kappa}$ conical singularities whose total angles around them are ${2\pi (k_i+1)}$, ${i=1,\dots,\kappa}$, then Alex Eskin, Maxim Kontsevich and Anton Zorich showed that the sum ${L(M)}$ of all non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle over ${SL(2,\mathbb{R})\cdot M}$ is

$\displaystyle L(M) = \frac{1}{12}\sum\limits_{i=1}^{\kappa}\frac{k_i(k_i+2)}{k_i+1} + \frac{1}{\# SL(2,\mathbb{Z})\cdot M} \sum\limits_{M_i\in SL(2,\mathbb{Z}) M}\sum\limits_{M_i=\cup \textrm{cyl}_{ij}} \frac{h_{ij}}{w_{ij}}$

where ${M_i=\cup\textrm{cyl}_{ij}}$ is the decomposition of ${M_i}$ into horizontal cylinders and ${h_{ij}}$, resp. ${w_{ij}}$ is the height, resp. width, of the horizontal cylinder ${\textrm{cyl}_{ij}}$.

In our setting, this formula gives

$\displaystyle L(\mathcal{O}) = \frac{1}{12}\frac{72\cdot 4\cdot 6}{5} + \frac{1}{15}\left(2\cdot 3\cdot 120\cdot\frac{1}{3} + 1\cdot 4\cdot 90\cdot \frac{1}{4} + 1\cdot 5\cdot 72\cdot \frac{1}{5}\right)$

and

$\displaystyle L(\mathcal{O}') = \frac{1}{12}\frac{72\cdot 4\cdot 6}{5} + \frac{1}{10}\left(1\cdot 1\cdot 180\cdot\frac{1}{2} + 1\cdot 4\cdot 90\cdot \frac{1}{4} + 1\cdot 5\cdot 72\cdot \frac{1}{5}\right)$

that is,

$\displaystyle L(\mathcal{O}) = \frac{278}{5} \neq 54 = L(\mathcal{O}')$

In particular, this allowed me to answer Jean-Christophe’s informal question: the knowledge of the commutator ${c=[\sigma,\tau]=[\sigma',\tau']}$ is not sufficient to determine the Lyapunov exponents.

3. Lyapunov exponents and T-systems?

Logically, this discussion led Jean-Christophe to ask a new question: how does the Lyapunov exponents of ${M(G,h,v)}$ relate to algebraic invariants of ${(G,h,v)}$? For example, is the Lyapunov exponent invariant’ equivalent to T-systems invariant’ (or is the Lyapunov exponent invariant’ a completely new invariant? [and, if so, can it be used to derive new interesting algebraic consequences for ${(G,h,v)}$?])

Remark 2 André Kappes and Martin Möller used Lyapunov exponents in another (but not completely unrelated) setting as new invariants’ to solve an algebraic problem (namely, classify commensurability classes of non-arithmetic lattices constructed by Pierre Deligne and George Mostow).

(Jean-Christophe and) I plan(ned) to come back to this question at some point in the future, but for now I will close this post here since I have nothing to say about this matter.