Given a group acting on a graph quasi-isometric to a tree, we give sufficient conditions for a pseudocharacter to be bushy. We relate this with the conditions studied by Bestvina and Fujiwara on their work on bounded cohomology and obtain some results on the space of pseudocharacters.
"Bushy pseudocharacters and group actions on quasitrees." Algebr. Geom. Topol. 12 (3) 1721 - 1739, 2012. https://doi.org/10.2140/agt.2012.12.1721