Geodesic currents and length compactness for automorphisms of free groups.

*(English)*Zbl 1166.20032Let \(F\) be a free group of finite rank \(k\geq 2\) and \(\vartheta F\) its hyperbolic boundary, which can be viewed as the space of right-infinite, freely reduced, words in an alphabet of \(F\). The boundary is endowed with the Cantor-set topology. Let \(\vartheta^2F=\{(x,y)\mid x,y\in\vartheta F\), \(x\neq y\}\). \(\vartheta^2F\) can be identified with the set of oriented bi-infinite geodesics in the Cayley graph of \(F\). The base-ball of \(\vartheta^2F\) is the set of geodesics passing through 1.

A geodesic current on \(F\) is a non-negative locally finite Borel measure on \(\vartheta^2F\) that is \(F\)-invariant. The length \(L(\eta)\) of a current \(\eta\) is the measure \(\eta(B)\) of the base-ball of \(\vartheta^2F\).

If \(\varphi\) is an automorphism of \(F\), then it is extended to a homeomorphism of \(\vartheta F\) (which is denoted again by \(\varphi\)) [D. Cooper, J. Algebra 111, 453-456 (1987; Zbl 0628.20029)]. This gives rise to an action of \(\varphi\) on the space of the geodesic currents on \(F\): Let \(\eta\) be a geodesic current, define \(\varphi\eta\) by setting \(\varphi\eta(S)=\eta(\varphi^{-1}(S))\) for a Borel subset \(S\subseteq\vartheta^2F\). Then it was proved [by I. Kapovich, Topological and asymptotic aspects of group theory. Contemp. Math. 394, 149-176 (2006; Zbl 1110.20034)] that \(\varphi\eta\) is a geodesic current on \(F\).

Let \(T\colon\vartheta F\to\vartheta F\) be the shift operator deleting the first letter of one right-infinite word. The space of frequency measures is the set of finite-mass \(T\)-invariant non-negative Borel measures on \(\vartheta F\).

There is a natural homeomorphism from the space of frequency measures to the space of geodesic currents (Theorem 3.4 in the paper or in [I. Kapovich, Exp. Math. 16, No. 1, 67-76 (2007; Zbl 1158.20014)]), therefore the action of an automorphism of \(F\) on the space of the geodesic currents on \(F\) induces an action of this automorphism on the frequency measures.

If we pass to the uniform current \(\eta_A\) (see Definition 3.5 in the paper), then the length of the automorphism \(\varphi\) is defined as the length of the image of the uniform current, that is, \(L(\varphi)=L(\varphi\eta_A)=\eta_A(\varphi^{-1}(B))\).

Now we can state the main result of the paper: For a sequence \(\varphi_n\) of automorphisms of \(F\), if there is a word \(w\) such that the cyclically reduced length of \(\varphi_n(w)\) goes to \(\infty\), then \(L(\varphi_n)\to\infty\).

A geodesic current on \(F\) is a non-negative locally finite Borel measure on \(\vartheta^2F\) that is \(F\)-invariant. The length \(L(\eta)\) of a current \(\eta\) is the measure \(\eta(B)\) of the base-ball of \(\vartheta^2F\).

If \(\varphi\) is an automorphism of \(F\), then it is extended to a homeomorphism of \(\vartheta F\) (which is denoted again by \(\varphi\)) [D. Cooper, J. Algebra 111, 453-456 (1987; Zbl 0628.20029)]. This gives rise to an action of \(\varphi\) on the space of the geodesic currents on \(F\): Let \(\eta\) be a geodesic current, define \(\varphi\eta\) by setting \(\varphi\eta(S)=\eta(\varphi^{-1}(S))\) for a Borel subset \(S\subseteq\vartheta^2F\). Then it was proved [by I. Kapovich, Topological and asymptotic aspects of group theory. Contemp. Math. 394, 149-176 (2006; Zbl 1110.20034)] that \(\varphi\eta\) is a geodesic current on \(F\).

Let \(T\colon\vartheta F\to\vartheta F\) be the shift operator deleting the first letter of one right-infinite word. The space of frequency measures is the set of finite-mass \(T\)-invariant non-negative Borel measures on \(\vartheta F\).

There is a natural homeomorphism from the space of frequency measures to the space of geodesic currents (Theorem 3.4 in the paper or in [I. Kapovich, Exp. Math. 16, No. 1, 67-76 (2007; Zbl 1158.20014)]), therefore the action of an automorphism of \(F\) on the space of the geodesic currents on \(F\) induces an action of this automorphism on the frequency measures.

If we pass to the uniform current \(\eta_A\) (see Definition 3.5 in the paper), then the length of the automorphism \(\varphi\) is defined as the length of the image of the uniform current, that is, \(L(\varphi)=L(\varphi\eta_A)=\eta_A(\varphi^{-1}(B))\).

Now we can state the main result of the paper: For a sequence \(\varphi_n\) of automorphisms of \(F\), if there is a word \(w\) such that the cyclically reduced length of \(\varphi_n(w)\) goes to \(\infty\), then \(L(\varphi_n)\to\infty\).

Reviewer: Dimitrios Varsos (Athenai)

##### MSC:

20F65 | Geometric group theory |

20E36 | Automorphisms of infinite groups |

20E05 | Free nonabelian groups |

##### References:

[1] | Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. · Zbl 0957.49001 |

[2] | Francis Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), no. 1, 71 – 158 (French). · Zbl 0671.57008 |

[3] | Daryl Cooper, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111 (1987), no. 2, 453 – 456. · Zbl 0628.20029 |

[4] | Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801 |

[5] | Ilya Kapovich, The frequency space of a free group, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 939 – 969. · Zbl 1110.20031 |

[6] | Ilya Kapovich, Clusters, currents, and Whitehead’s algorithm, Experiment. Math. 16 (2007), no. 1, 67 – 76. · Zbl 1158.20014 |

[7] | Ilya Kapovich, Currents on free groups, Topological and asymptotic aspects of group theory, Contemp. Math., vol. 394, Amer. Math. Soc., Providence, RI, 2006, pp. 149 – 176. · Zbl 1110.20034 |

[8] | Vadim Kaimanovich, Ilya Kapovich, and Paul Shupp. The subadditive ergodic theorem and generic stretching factors for free group automorphisms. Preprint, arXiv:math.GR/0504105, to appear in Israel J. Math., 2005. · Zbl 1173.20031 |

[9] | Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. · Zbl 0368.20023 |

[10] | R. Martin. Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups. Ph.D. thesis, University of California, Los Angeles, 1995. |

[11] | J. H. C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. (2) 37 (1936), no. 4, 782 – 800. · Zbl 0015.24804 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.