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The automorphism group of the free group of rank 2 is a \(\text{CAT}(0)\) group. (English) Zbl 1205.20050

The authors define a finitely generated group \(G\) to be a \(\text{CAT}(0)\) group if \(G\) acts cocompactly and properly discontinuously by isometries on a \(\text{CAT}(0)\) metric space \(X\). Such an action is called geometric.
The main resut of the paper is that \(\operatorname{Aut}(F_2)\) is a \(\text{CAT}(0)\) group, where \(F_2\) is the free group on 2 generators.
To prove this, the authors use the following: 1) \(\operatorname{Aut}(F_2)\) is isomorphic to \(\operatorname{Aut}(B_4)\), where \(B_4\) is the braid group on 4 generators. 2) \(\text{Inn}(B_4)\) has index two in \(\operatorname{Aut}(B_4)\). 3) A result due to T. Brady [Mich. Math. J. 47, No. 2, 313-324 (2000; Zbl 0996.20022)] that shows that \(B_4\) acts faithfully and geometrically on a \(\text{CAT}(0)\) 3-complex.
It follows from the above facts that \(\text{Inn}(B_4)\) acts faithfully and geometrically on a \(\text{CAT}(0)\) 2-complex \(X_0\). The authors then exhibit an extra isometry of \(X_0\) which extends the faithful geometric action of \(\text{Inn}(B_4)\) to a faithful geometric action of \(\operatorname{Aut}(B_4)\). The last step constitutes the bulk of the paper.
Reviewer: Mahan Mj (Howrah)

MSC:

20F65 Geometric group theory
20F28 Automorphism groups of groups
20E05 Free nonabelian groups
57M07 Topological methods in group theory

Citations:

Zbl 0996.20022
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References:

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