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.

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