Characterizing rigid simplicial actions on trees

We extend Forester’s rigidity theorem so as to give a complete characterization of rigid group actions on trees (an action is rigid if it is the only reduced action in its deformation space, in particular it is invariant under automorphisms preserving the set of elliptic subgroups). Let T be a simplicial tree with a cocompact action of a group G (i.e. the BassSerre tree associated to a decomposition of G as a finite graph of groups). A G-tree T ′ is a deformation of T if it may be obtained from T by a finite sequence of expansions and collapses (elementary moves coming from the canonical isomorphism A∗B B ≃ A). These moves do not change the set of elliptic subgroups (a subgroup is elliptic if it fixes a point in the tree). Conversely, Forester proved [3] that any cocompact G-tree T ′ with the same elliptic subgroups as T is a deformation of T . Since an expansion makes the tree more complicated, and a collapse makes it simpler, it is natural to restrict to reduced trees. A tree T is reduced if one cannot perform a collapse on T . Equivalently, T is reduced if, whenever an edge e has the same stabilizer as one of its endpoints, then both endpoints of e are in the same G-orbit (i.e. e projects onto a loop in the quotient graph). The tree T is rigid if it is reduced, and it is the only reduced tree in its deformation space (up to equivariant isomorphism). In other words, all deformations of T (trees with the same elliptic subgroups as T ) may be reduced to T by collapse moves. Rigidity provides a canonical element Tred in the deformation space; in particular, any automorphism of G that preserves the set of elliptic subgroups leaves Tred invariant (see [1, 4, 5, 7, 8, 9] for examples and applications to JSJ splittings and automorphisms). Forester proved that “strongly-slide-free” trees are rigid ([3], see also [6]). The purpose of this note is to extend Forester’s theorem. Our extension is optimal: we obtain a complete characterization of rigid trees. Before stating our result, let us illustrate it on generalized Baumslag-Solitar groups [2, 9]. Note that these groups have been classified up to quasi-isometry [10]. The rigidity we are studying here is not quasi-isometric rigidity of groups, as the group is fixed once and for all. We consider a finite graph of groups with each vertex and edge group isomorphic to Z. It is pictured as a labelled graph, each label being the index of the edge Typeset by AMS-TEX 1 group in the vertex group (see figure 1). [One should allow negative labels, but we will not bother here.]