- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:00
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
8
2014
2
On the geometry of the edge splitting complex
Lucas
Sabalka
Saint Louis University, St Louis, UNITED STATES
Dmytro
Savchuk
University of South Florida, TAMPA, UNITED STATES
Outer automorphisms of free groups, curve complex, quasi-isometry, hyperbolicity, asymptotic dimension
The group Out of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which Out acts, in analogy with the curve complex for the mapping class group. Here, we focus on one of these proposed analogues: the edge splitting complex $\mathcal{E}\mathcal{S}_n$, equivalently known as the separating sphere complex. We characterize geodesic paths in its 1-skeleton $\mathcal{E}\mathcal{S}_n^1$ algebraically, and use our characterization to find lower bounds on distances between points in this graph. Our distance calculations allow us to find quasiflats of arbitrary dimension in $\mathcal{E}\mathcal{S}_n$. This shows that $\mathcal{E}\mathcal{S}_n$: is not hyperbolic, has infinite asymptotic dimension, and is such that every asymptotic cone is infinite dimensional. These quasiflats contain an unbounded orbit of a reducible element of Out. As a consequence, there is no coarsely Out-equivariant quasiisometry between $\mathcal{E}\mathcal{S}_n$ and other proposed curve complex analogues, including the regular free splitting complex $\mathcal{F}\!\mathcal{S}_n$, the (nontrivial intersection) free factorization complex $\mathcal{F\!F}_n$, and the free factor complex $\mathcal{F}_n$.
Group theory and generalizations
565
598
10.4171/GGD/239
http://www.ems-ph.org/doi/10.4171/GGD/239