Low complexity among principal fully irreducible elements of Out($F_3$)

TL;DR

The paper identifies the minimal stretch factor realized by a fully irreducible element of $\mathrm{Out}(F_3)$ and shows it is attained by a principal fully irreducible. It constructs a rank-3 principal stratum automaton and an explicit single-fold train-track map $\mathfrak{g}$ whose PF eigenvalue equals the smallest possible growth, namely the real root of $x^5-x-1$. The authors prove $\mathfrak{g}$ is ageometric and principal, compute its transition matrix, and rule out a competing degree-5 Perron number by a trace argument, establishing minimality. They also prove, up to edge relabeling, that this single-fold principal representative is unique in any rank, with nonexistence of such maps for $r\ge 4$. Overall, the work connects fold theory, Perron numbers, and principal axes in Out($F_r$), clarifying the structure of minimal-growth principal fully irreducibles on rank-3 Outer space.

Abstract

We find the shortest realized stretch factor for a fully irreducible $\varphi\in\mathrm{Out}(F_3)$ and show that it is realized by a "principal" fully irreducible element. We also show that it is the only principal fully irreducible produced by a single fold in any rank.

Figures (4)

• Figure 1: The right-hand image is the ltt structure $G(\mathfrak{g})$ for the map $\mathfrak{g}$ of (\ref{['g']}) with the taken turns as in (\ref{['turns']}). The 3 local Whitehead graphs are colored, with the stable Whitehead graphs colored purple (missing only $\bar{c}$). The ideal Whitehead graph is the union of the purple graphs, i.e. of the stable Whitehead graphs.
• Figure 2: The Stallings fold decomposition of $\mathfrak{g}$ is a fold $f$, then homeomorphism $g_{\sigma}$. This map is indicated in dotted gold in \ref{['f:mA3']}.
• Figure 3: This figure depicts the Rank-3 Principal Stratum Automaton. Permutations of the automata are in green; note that if moving right to left across the page one must read the permuatation right to left as well. Compositions of included permutations, as well as color-preserving graph symmetries, are implicitly included. (Symmetries are included in blue at II as an example.) The map of Figure \ref{['f:map3']} is in dotted gold.
• Figure :

Theorems & Definitions (19)

• Proposition 2.1: Full Irreducibility Criterion, IWGII
• Definition 3.1: Graph/permutation relabeling
• Lemma 3.2: Lonely Direction Lemma
• proof
• Definition 3.3: Rank-3 Principal Stratum Automaton $\widehat{\mathcal{A}_3}$
• Theorem C
• proof
• Proposition 3.4
• proof
• Lemma 4.1