Symmetric automorphisms of free products

Proper Actions of Automorphism Groups of Free Products of Finite Groups

“…For each i, choose a λ i ∈ G i − {1}. Let P(J 0 ) be the Whitehead poset constructed in [10]. Elements of P(J 0 ) correspond to labelled bipartite trees, where the n + 1 labels come from the set { * , G 1 , .…”

“…The poset structure in P(J 0 ) comes from an operation called folding (when the elements are thought of as labelled trees) or by setting A(k) ≤ B(k) if elements of A(k) are unions of elements of B(k). See McCullough and Miller [10] for more details.…”

“…Observe that elements of P (J ) correspond to pointed trees with labels in J , and that P (J ) is (n − 1)-dimensional. Denote elements of P (J ) as pairs (J , A) as in [10]. Now mimic the construction in section 2 of [10].…”

“…In this paper, we will assume that each G i is finite so that Aut(G) = ΣAut(G) by the Kurosh subgroup theorem. We will construct an EΣAut 1 (G)-space L(G) based on McCullough-Miller's [10] space of trees, which uses rooted trees similar to those found in Gutiérrez-Krstić [6]. Here ΣAut 1 (G) is the kernel of the projection ΣAut(G) → Σ n .…”

“…Gilbert [5] further refines these methods and gives a presentation for ΣAut(G). In [10], McCullough and Miller provide a comprehensive work about symmetric automorphism groups of free products and define McCullough-Miller space. In [3] Bridson and Miller show that every finite subgroup of ΣAut 1 (G) fixes a point of McCullough-Miller space K 0 (G).…”