Actions of Automorphism Groups on Real Trees

Alexandre Ivanov

Let G be an infinite group acting on an R-tree by homeomorphisms. The action is called non-nesting if no g ∈ G maps an arc properly into itself [G.Levitt, Non-nesting actions on real trees, Bull. London Math. Soc. 30(1998), 46 – 54]. In this abstract we always assume that G is a closed subgroup of Sym(ω) (the automorphism group of a structure M on ω) and its actions are non-nesting with closed stabilizers (the stabilizer is closed when the action is Borel).

A structure N realizes orthogonal copies if for any tuple b̄ ⊂ N and any finite A ⊂ N there exists b̄′ ⊂ N of the type tp(b̄) such that the restriction map tp(ā/b̄′) → tp(ā), with ā ⊆ A, is one-to-one [A.Ivanov, Finite covers, cohomology and homogeneous structures, Proc. London Math. Soc., 78(1999), 1 – 28].

Theorem 1. Let M be a countable ω-homogeneous sructure realising orthogonal copies and G = Aut(M). Then for any non-nesting action of G on a real tree with clopen stabilizer any g ∈ G fixes a point.

The theorem implies that if M additionally has the small index property then each g ∈ G fixes a point with respect to any action on a Z-tree. This implies that the automorphism group of an arithmeticaly saturated model of PA has property FA defined in [J.P.Serre. Trees. Springer-Verlag 1980].

Theorem 2. Let A(Q) be the group of all order-preserving permutations of (Q, <). Let G ∈ {A(Q), Sym(ω)}. Then for any non-nesting action of G on a real tree each element of G fixes a point.

Let A be a countable periodic abelian group and H(A) be the universal locally finite extension of A [K.Hickin, Universal locally finite central extensions of groups, Proc. London Math. Soc., 52(1986), 53 – 72].

Theorem 3. For any isometric action of G = Aut(H(A)) on a real tree the group G acts by translations on a G-invariant geodesic line or any element of G fixes a point. The group G has property FA if and only if for no p the p-component of the divisible part of A consists of finitely many quasicyclic groups.

Address: Institute of Mathematics, Wrocław University, pl.Grunwaldzki 2/4, 50-384 Wrocław, Poland

E-mail: ivanov@math.uni.wroc.pl