Title: Algebraic laminations for free products and arational trees,
Abstract: This work is the first step towards a description of the Gromov boundary of
the free factor graph of a free product, with applications to subgroup
classification for outer automorphisms. We extend the theory of algebraic
laminations dual to trees, as developed by Coulbois, Hilion, Lustig and
Reynolds, to the context of free products; this also gives us an opportunity to
give a unified account of this theory. We first show that any $\mathbb{R}$-tree
with dense orbits in the boundary of the corresponding outer space can be
reconstructed as a quotient of the boundary of the group by its dual
lamination. We then describe the dual lamination in terms of a band complex on
compact $\mathbb{R}$-trees (generalizing Coulbois-Hilion-Lustig's compact
heart), and we analyze this band complex using versions of the Rips machine and
of the Rauzy-Veech induction. An important output of the theory is that the
above map from the boundary of the group to the $\mathbb{R}$-tree is 2-to-1
almost everywhere.
A key point for our intended application is a unique duality result for
arational trees. It says that if two trees have a leaf in common in their dual
laminations, and if one of the trees is arational and relatively free, then
they are equivariantly homeomorphic.
This statement is an analogue of a result in the free group saying that if
two trees are dual to a common current and one of the trees is free arational,
then the two trees are equivariantly homeomorphic. However, we notice that in
the setting of free products, the continuity of the pairing between trees and
currents fails. For this reason, in all this paper, we work with laminations
rather than with currents.