The geometry of the cancellation metric on free groups

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Abstract: The cancellation length of a word is the smallest number of letters to delete to obtain a word representing the trivial element. It well defines a norm on free groups and the associated metric is called the cancellation metric. Equivalently, it is the word metric associated with the set consisting of all generators, their inverses and conjugates. The purpose of the talk is to introduce this metric, present some basic properties as well as some non-trivial and surprising ones. Examples of results (let $m,m ? 2$): Algebraic rigidity: A homomorphism $F_m ? F_n$ is a quasi-isometry if and only if it is an isomorphism. Quasi-algebraic flexibility: There exists a quasi-homomorphism $F_m ? F_n$ which is a quasi-isometry. Torsion in the cone: There exist elements $g \in F_2$ such that $\|g^2\|$ is much smaller than $?g?$. Joint work with Assaf Libman, Micha? Marcinkowski and Zipei Nie.