The maximal number of bushy free factors that coarsely embed into a standard HHS is the orthogonality number of its bushy domains; this distinguishes Torelli groups, Johnson kernels and surface braid groups.
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Coarse embeddings of products of trees as quasi-isometry invariants
The maximal number of bushy free factors that coarsely embed into a standard HHS is the orthogonality number of its bushy domains; this distinguishes Torelli groups, Johnson kernels and surface braid groups.