Submanifold projections and hyperbolicity in ${\rm Out}(F_n)$

Abstract

The free splitting graph of a free group $F_n$ with $n\geq 2$ generators is a hyperbolic ${\rm Out}(F_n)$-graph which has a geometric realization as a sphere graph in the connected sum of $n$ copies of $S^1\times S^2$. We use this realization to construct submanifold projections of the free splitting graph into the free splitting graphs of proper free factors. This is used to construct for $n\geq 3$ a new hyperbolic ${\rm Out}(F_n)$-graph. If $n=3$, then every exponentially growing element acts on this graph with positive translation length.

Theorems & Definitions (60)

• Theorem 1
• Theorem 2: Theorem 5.1 of BF14b
• Definition 2.1
• Proposition 2.2
• proof
• Example 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof