A Generalization of Culler’s Theorem

Culler's theorem states that for a finitely generated free group F, of rank at least 2, any finite subgroup of Out(F) fixes a point in Outer Space. This theorem is comparable to Nieslen Realization: for a closed surface with negative Euler characteristic, any finite subgroup of the mapping class group fixes a point in the Teichmüller sapce for the surface as proved by Kerckhoff. For nonnegative integers n, we define a class of groups G(n) and prove a similar statement for their outer automorphism groups. For a closed surface with negative Euler characteristic Σ, the mapping class group MCG(Σ) acts on Teichmüller space, the space of hyperbolic metrics on Σ. The stabilizers of this action are finite subgroups of MCG(Σ). Kerckhoff [15] proved the converse, namely any finite subgroup of MCG(Σ) fixes a point in T Σ. This result is known as Nielsen Realization; Nielsen and others had shown the result for various special cases. In a similar manner, for a free group of rank n ≥ 2, the outer automorphism group Out(F n) acts on Outer Space. The stabilizers of this action are finite subgroups of Out(F n) and Culler [5] proved that any finite subgroup of Out(F n) fixes some point in Outer Space. Both Teichmüller space and Outer Space are contractible [7]. For a nonnegative integer n we introduce a class of groups denoted G(n), where the outer au-tomorphism group of any group in this class has a similar realization statement. In other words, for every group G ∈ G(n), there is a contractible space on which Out(G) acts and we are able to determine that certain subgroups of Out(G) related to stabilizers are indeed stabilizers themselves. For n = 0, 1 we show that any group which is commensurable to a subgroup of a stabilizer actually fixes a point (Corollary 5.2). Recall that two subgroups are commensurable if they share a finite index subgroup. The class G(0) is the class of virtually finitely generated free groups of rank at least 2, thus our result is a generalization of the above mentioned theorem of Culler. In general, we are only able to show that subgroups of Out(G) commensurable to polycyclic subgroups of stabilizers actually fix a point. We define G 0 (n) as the class of groups which act on a locally finite simplicial tree without an invariant point or line, such that the edge stabilizers are …