Abstract: Let
F be a non-abelian free group. Given a word w ∈ F, its cancellation
length c(w) is the minimal number of letters to be removed from w to
obtain a word representing the trivial element of F. The cancellation
length is constant on conjugacy classes and hence defines a bi-invariant
metric d on F. I will present what I know about the geometry of (F,d).
First of all, d is a word metric associated with the generating set
consisting of the standard generators of F together with all their
conjugates. Hence, we can consider the associated Cayley graph, which
contains interesting subgraphs. For example, trees, 1-skeleta of CAT(0)
cube complexes, 1-skeleta of complexes of injective words and others.
The group structure of F exhibits peculiar behaviour with respect to the
cancellation length. For example, there are elements w ∈ F with c(w²) ≤
c(w). The shortest one for which the equality holds is a word of length
14. The shortest w (known to me) for which the strict inequality holds
is ridiculously long. As a consequence we get that the asymptotic cone
of (F,d) has 2-torsion (asymptotic cones of groups equipped with
bi-invariant metrics are metric groups).
The cancellation length is not that abstract as it looks. For example,
it is related to the minimal area of a planar disc bounded by a closed
planar curve (I will say more on that). Moreover, a word in F₂ can be
thought of as an RNA chain. In nature RNA chains sometimes fold and form
hairpin-shaped structures. Biologists interested in the RNA folding
discovered the above metric in 1999 and an effective algorithm that
computes the cancellation length of the element corresponding to an RNA
chain.
