Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected metric graph of total edge-length $1$ with no valence $1$ or $2$ vertices and no separating edges and $\mu\colon F_n \rightarrow \pi_1(X_n)$ is the conjugacy class of an isomorphism.
The space $X_n$ was introduced in the paper
M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91–119.
The main theorem of this paper is that $X_n$ is contractible. They prove this in the following way. Let $W_0$ be the following set of elements of $F_n$:
- $x_i$ with $1 \leq i \leq n$, and
- $x_i x_j$ and $x_i x_j^{-1}$ with $1 \leq i < j \leq n$.
For each $w \in W_0$, and each point $(G,\mu)$ of $X_n$, let $|(G,\mu)|_w$ be the length of the shortest unbased loop in $G$ representing the conjugacy class $w$. Define
$$|(G,\mu)| = \sum_{w \in W_0} |(G,\mu)|_w.$$
They use this as a sort of "Morse function" on $X_n$, and study its level sets. One of their key results is as follows. A rose in $X_n$ is a point $(R,\mu)$, where $R$ is a metric graph with $1$ vertex and $n$ edges, each of length $1/n$. Let $(R_0,\mu_0)$ be the rose where $\mu_0$ identifies the loops in $R_0$ with the basis elements $x_1,\ldots,x_n$.
Proposition (Prop 6.2.5 in the above paper): The rose $(R_0,\mu_0)$ is the unique minimum of the above Morse function among roses, i.e., if $(R,\mu)$ is another rose then $|(R_0,\mu_0)| \leq |(R,\mu)|$ with equality if and only if $(R,\mu) = (R_0,\mu_0)$.
I remark that Prop 6.2.5 has the opposite inequality to the one I wrote, but I think this is a typo and the proof seems to give the above. The proof of this proposition is pretty simple and convincing.
However, it seems to be contradicted in the following paper:
J. Smillie and K. Vogtmann, Length functions and outer space, Michigan Math J. 39 (1992) 485–493.
In fact, Theorem 1 of this paper seems to say that not only is the above proposition false, but it remains false if $W_0$ is replaced by any other finite set of elements of $F_n$. The authors of this paper do not say anything about this contradicting anything in the previous paper (even though they share an author, so presumably if I am right they were aware of this).
Question: Am I misinterpreting any of these papers? If not, what exactly is going on here?
