Manuel Ladra, Pedro V Silva, and Enric Ventura (2012)

# Bounding the gap between a free group (outer) automorphism and its inverse

arXiv:1212.6749.

Two complexity functions \$alpha\_r\$ and \$beta\_r\$ are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of \$F\_r\$ and the norm of its inverse. The exact complexity of \$alpha\_2\$ and \$beta\_2\$ is computed. For rank \$r geqslant 3\$, polynomial lower bounds are provided for \$alpha\_r\$ and \$beta\_r\$, and the existence of a polynomial upper bounded is proved for \$beta\_r\$.

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