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Gaboriau-Jaeger-Levitt-Lustig (Theorem II.1) constructed an invariant $\mathbb{R}$-tree $T$ with $F_n$ action given any outer automorphism of free groups $\Phi\in \mathrm{Out}(F_n)$. They showed that $F_n$ acts on $T$ isometrically, non-trivially, minimally with trivial arc stabilizers. The invariant tree being non-simplicial with dense orbits follows from a quick argument of Proposition 3.10 in Paulin 1997. I want to take a look at a simple example of what this construction gives us. But I got something simplicial and I'm confused.

My example: consider the non-abelian free group $F_2=\langle a,b\rangle$ and an irreducible element $\Phi\in\mathrm{Out}(F_2)$ be $a\mapsto b,b\mapsto ab$. Let $R_2$ be a rose of rank $2$ where we label $a,b$ clockwise on the two circles from left to right. Consider the train track representative $\varphi:R_2\to R_2$ mapping $a\mapsto b$ and $b\mapsto ab$ and the transition matrix $M(\varphi)$ that has Perron-Frobenius eigenvalue $\lambda=\frac{1+\sqrt{5}}{2}$, left Perron-Frobenius eigenvector $L=[\frac{\lambda^2}{1+\lambda^2},\frac{\lambda^3}{1+\lambda^2}]$, and right Perron-Frobenius eigenvector $R=[\lambda^{-2},\lambda^{-1}]^T$ satisfying $\sum_i R_i=1$ and $L\cdot R=1$.

Equip $R_2$ the Perron-Frobenius metric $L$ such that $\ell(a)=L(1),\ell(b)=L(2)$. Let $(T,d)$ be the universal cover of $R_2$ with the lifted metric and denote $||\cdot||_{T}$ its length. We construct the invariant tree $T_+:=(T,d^*_{+})$ where

$$d_+(x,y)=\lim_{p\to\infty} \frac{d(\varphi^p(x),\varphi^p(y))}{\lambda^p}$$

and $d^*_{+}$ is the induced metric from the pseudo metric $d_{+}$. For any segment $\beta$ in $T$, we have $d_+(\beta)=\lim_{p\to\infty}\frac{||\varphi^p(\beta)||_{T}}{\lambda^p}$. Since $\varphi$ is a train track map then $d_+(\beta)=||\beta||_{T}$ for any legal path $\beta$ in $T$. For arbitrary path $\beta$ in $T$, we have the estimation that

$$\sum_{e}\frac{N_e^p}{\lambda^p}\cdot\frac{||\varphi^{p_0}(e)||_T}{\lambda^{p_0}}[1-\frac{2BBT(\varphi)}{||\varphi^{p_0}(e)||_T}]\leq\frac{||\varphi^{p+p_0}(\beta)||}{\lambda^{p+p_0}}\leq \sum_{e}\frac{N_e^p}{\lambda^p}\cdot\frac{||\varphi^{p_0}(e)||_T}{\lambda^{p_0}}$$

where $N_e^p$ denotes the occurrence of $e$ in $\varphi^p(\beta)$. When $p_0$ is large, it follows from Perron-Frobenius theory that $d_+(\beta)$ is approximately $||\beta||_T\cdot \sum_{e}R(e)||e||=||\beta||_T$. So, it turns out $d_+(\beta)=d(\beta)$ for any edge path $\beta$ in $T$. But how is this an $\mathbb{R}$-tree with dense orbits?

Thanks for your help!

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  • $\begingroup$ Your wrote the formula $$a \mapsto b \quad b \mapsto ab$$ for $\Phi$, but then a different formula $$a \mapsto a \quad b \mapsto ab$$ for $\phi$. $\endgroup$ Commented Nov 26 at 2:44
  • $\begingroup$ Also, what definition of "minimality" of the action are you using? And what theorem are you citing in your first two sentences? $\endgroup$ Commented Nov 26 at 2:51
  • $\begingroup$ Hi, @LeeMosher thanks for your response! I've fixed my typo and add the citation. By minimality, I mean there is no proper $G$-invariant subtree. $\endgroup$ Commented Nov 26 at 17:45
  • $\begingroup$ Okay then, next question. What is the exact citation of the Paulin argument? It certainly is NOT true that every isometric, non-trivial, minimal, isometric action of $F_n$ on a tree with trivial arc stabilizers has dense orbits. Think of the deck action of $F_n$ on the universal cover of the standard rank $n$ rose. $\endgroup$ Commented Nov 26 at 18:38
  • $\begingroup$ I have also added it (Proposition 3.10) in the question, but it's a French paper! The fact is that to have dense orbits, we need minimalily and an $F_n$-equivariant homothety i.e. $H:T\to T$ such that $d(Hx,Hy)=\lambda d(x,y)$ for $\lambda>1$ and $Hg=\Phi(g)H$. The argument goes as follows: pick $x\in T$ and a ball of raidus $r$ that doesn't intersect the orbit of $y$, then the ball centered at $H^nx$ of radius $\lambda^n r$ doesn't intersect the orbit of $H^ny$. But non-triviality and minimality tells us we have a compact fundamental domain which is a contradiction. $\endgroup$ Commented Nov 26 at 19:23

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This is more of a long comment. Your statement of Theorem 2.1 in your post is missing an important part of the conclusion of that theorem, regarding a homothety on $T$. You do give an incomplete account of that conclusion in your comments, but to answer your question a correct and complete account is needed. Namely, the $\mathbb R$-tree $T$ (equipped with a minimal isometric $F_n$ action) carries some additional structure:

There exists $\lambda \ge 1$ and $H : T \to T$ such that $H$ is a homeomorphism, and $H$ is a $\lambda$-homothety (meaning that $d(H(x),H(y)) = \lambda H(x,y)$ for all $x,y \in T$), and $H$ satisfies a certain equivariance property with respect to $\Phi$ (the exact form of which is not particularly relevant to your question).

You also cite a theorem of Paulin which says that IF $\lambda > 1$ then there is an additional conclusion saying the action of $F_n$ on $T$ has dense orbits.

However, if $\lambda=1$ then this additional conclusion need not hold.

Regarding your example, I can't really follow your estimates; I don't know what you mean by BBT, nor what role $p_0$ plays. But I'll make a couple of points that might help you analyze your own work.

There exist paths $\beta$ in $R_2$ with arbitrarily large value of $d(\beta)$ but $d_+(\beta)=0$; this happens because for some $k$ the path $\phi^k(\beta)$ is path homotopic to a constant. A pathlike this is called "pretrivial". What does your method say about pretrivial paths?

There also exist bi-infinite paths which can be written as an increasing union of pretrivial paths. Examples include leaves of the repelling lamination realized in $T$. What does your method say about these lines?

There is a general pattern that you can find in one of the train track papers, which will hold for your example: for any path $\beta$, letting $\phi^k_\#(\beta)$ be the path obtained by pulling $\phi^k(\beta)$ tight, the length of $\phi^k_\#(\beta)$ shrinks exponentially for a while by factors close to $\mu > 1$ (which is the expansion factor for $\phi^{-1}$), and then switches modes and grows exponentially by factors getting asymptotically close to $\lambda$. (This is somewhat like the behavior of the length of a sequence of vector $T^k(v)$ where $T$ is a Perron-Frobenius matrix and $v$ is a vector). Predicting a priori at what value of $k$ the switch occurs between exponential shrinkage and exponential growth is tricky. I suspect that where you went wrong is in not taking this delicate switch into account.

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  • $\begingroup$ Thanks so much for your response! I was wrong that the argument only works for legal paths $\beta$ in $T$! For example, $ab^{-1}$ contains an illegal turn where $d_+(ab^{-1})=\frac{d_+(a^{-1})}{\lambda}$. The BBT represents for the bounded cancellation number (or bounded backtracking) and $p_0$ is a large constant such that BBT($\varphi$) is small. This is Lemma 7.1 of Levitt-Lustig "Irreducible automorphism of $F_n$ have north-south dyanmics on compactified outer space". $\endgroup$ Commented 7 hours ago

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