The boundary of outer space in rank two.

*(English)*Zbl 0786.57002
Arboreal group theory, Proc. Workshop, Berkeley/CA (USA) 1988, Publ., Math. Sci. Res. Inst. 19, 189-230 (1991).

[For the entire collection see Zbl 0744.00026.]

In a previous paper the authors introduced a space \(X_ n\), now known as outer space, on which the group \(\text{Out}(F_ n)\) of outer automorphisms of a free group of rank \(n\) acts virtually freely [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)]. The authors try to develop an analogy between the action of \(\text{Out}(F_ n)\) on outer space and the action of the mapping class group of a surface on the Teichmüller space of that surface. In particular, the action of the mapping class group on Teichmüller space was exploited by Thurston in his classification of automorphisms of surfaces. Thurston gives an embedding of the Teichmüller space into an infinite dimensional projective space and shows that its closure in this projective space is a finite-dimensional ball; he then uses the fact that the ball has the fixed point property to analyze the action of a single automorphism on the closure of Teichmüller space. The authors aim to adapt Thurston’s theory to automorphisms of free groups. It was shown in the above mentioned paper (l.c.) that the outer space \(X_ n\) is contractible of dimension \(3n-4\), and also by the first author and J. W. Morgan [Proc. Lond. Math. Soc., III. Ser. 55, 571-604 (1987; Zbl 0658.20021)] that its closure \(\overline {X}_ n\), in the infinite dimensional projective space \(P^ C= (\mathbb{R}^ C- \{0\})/\mathbb{R}^*\) where \(C\) is the set of conjugacy classes of \(F_ n\), is compact. In this paper the authors restrict themselves to the case \(n=2\). They give an explicit description of the closure \(\overline {X}= \overline {X}_ 2\) of outer space of rank 2. In particular they show that \(\overline {X}\) is contractible, and give an imbedding of \(\overline {X}\) as a two-dimensional subset of \(\mathbb{R}^ 3\) which makes it clear that \(\overline {X}\) is an ANR.

In a previous paper the authors introduced a space \(X_ n\), now known as outer space, on which the group \(\text{Out}(F_ n)\) of outer automorphisms of a free group of rank \(n\) acts virtually freely [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)]. The authors try to develop an analogy between the action of \(\text{Out}(F_ n)\) on outer space and the action of the mapping class group of a surface on the Teichmüller space of that surface. In particular, the action of the mapping class group on Teichmüller space was exploited by Thurston in his classification of automorphisms of surfaces. Thurston gives an embedding of the Teichmüller space into an infinite dimensional projective space and shows that its closure in this projective space is a finite-dimensional ball; he then uses the fact that the ball has the fixed point property to analyze the action of a single automorphism on the closure of Teichmüller space. The authors aim to adapt Thurston’s theory to automorphisms of free groups. It was shown in the above mentioned paper (l.c.) that the outer space \(X_ n\) is contractible of dimension \(3n-4\), and also by the first author and J. W. Morgan [Proc. Lond. Math. Soc., III. Ser. 55, 571-604 (1987; Zbl 0658.20021)] that its closure \(\overline {X}_ n\), in the infinite dimensional projective space \(P^ C= (\mathbb{R}^ C- \{0\})/\mathbb{R}^*\) where \(C\) is the set of conjugacy classes of \(F_ n\), is compact. In this paper the authors restrict themselves to the case \(n=2\). They give an explicit description of the closure \(\overline {X}= \overline {X}_ 2\) of outer space of rank 2. In particular they show that \(\overline {X}\) is contractible, and give an imbedding of \(\overline {X}\) as a two-dimensional subset of \(\mathbb{R}^ 3\) which makes it clear that \(\overline {X}\) is an ANR.

Reviewer: S.Andreadakis (Athens)

##### MSC:

57M07 | Topological methods in group theory |

20E08 | Groups acting on trees |

20E05 | Free nonabelian groups |

20E36 | Automorphisms of infinite groups |

20F65 | Geometric group theory |