Presentations of the Automorphism Group of a Free Product

Proper Actions of Automorphism Groups of Free Products of Finite Groups

“…Finally, if W is a set of words in G, we can constructW by sending w ∈ W tow = wλ n+1 ∈W (cf. Proposition 2.18 in [5], which is the basic idea of what we are doing here in this adjustment.) Then |w| v + 1 = |w|v for any w ∈ W so that red W (α, v) = redW (ᾱ,v) (see [10] for a definition of the reductivity red of a Whitehead move) for any α ∈ ΣAut(G).…”

“…In [4] Collins and Zieschang establish the peak reduction methods that underly all of the contractibility arguments here. 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.…”

Mapping-Class Groups of 3-Manifolds in Canonical Quantum Gravity

“…We can then determine a finite presentation of G F (Σ) by means of the so-called Fouxe-Rabinovitch presentation for the automorphism group of a free product of groups developed in [16,17]; see also [63] and [22]. Let generally…”

“…In that case we have found all generators after adjoining these additional n s + m generators. A complete list of relations can then be found from the Fouxe-Rabinovitch relations for Aut(G) = G 1 * · · · G n+m (see Chapter 5.1 of [63]) 22 and some added relations which the n s + m added generators have to satisfy. The latter are not difficult to find due to the simple geometric interpretation of the diffeomorphisms of Fig.…”

“…That is, the slide conjugates [γ] with [ℓ]. 22 The original papers by Fouxe-Rabinovitch [17,16] contained some errors in the relations which were corrected in [63]; see also [22]. 23 The added generators are internal transformations (i.e.…”