an:06406277
Zbl 1325.20035
Kapovich, Ilya; Lustig, Martin
Cannon-Thurston fibers for iwip automorphisms of \(F_N\)
EN
J. Lond. Math. Soc., II. Ser. 91, No. 1, 203-224 (2015).
00341644
2015
j
20F65 20F67 20E36 20E05 57M07 37B10 57M50
word-hyperbolic groups; Cannon-Thurston map; iwip automorphisms; fully irreducible automorphisms; mapping torus groups
According to the notion of a Cannon-Thurston map studied by \textit{J. W. Cannon} and \textit{W. P. Thurston}, [Geom. Topol. 11, 1315-1355 (2007; Zbl 1136.57009)], in group-theoretic terms an analogous notion is developed by \textit{M. Mitra} [in Geom. Funct. Anal. 7, No. 2, 379-402 (1997; Zbl 0880.57001); Topology 37, No. 3, 527-538 (1998; Zbl 0907.20038); Geom. Topol. Monogr. 1, 341-364 (1998; Zbl 0914.20034)].
If \(G\) is a word-hyperbolic group and \(H\) a word-hyperbolic subgroup, and if the inclusion \(\iota\colon H\to G\) extends to a continuous map \(\widehat\iota\colon\partial H\to\partial G\), then the map \(\widehat\iota\) is called the Cannon-Thurston map. In particular if the Cannon-Thurston map exists, then it is unique. It is well known that if \(H\leq G\) is a quasiconvex subgroup of a word-hyperbolic group \(G\), then \(H\) is word-hyperbolic and the inclusion extends to a continuous topological embedding \(\partial H\to\partial G\). Thus in this case the Cannon-Thurston map exists and, moreover, is injective. A result of Mitra [in loc. cit., Zbl 0907.20038] states that whenever \(1\to H\to G\to Q\to 1\) is a short exact sequence of word-hyperbolic groups, then the inclusion \(H\leq G\) extends to a continuous Cannon-Thurston map \(\widehat\iota\colon\partial H\to\partial G\). Until recently, it has been unknown whether there are any inclusions \(H\leq G\) (with \(H\) and \(G\) word-hyperbolic) where the Cannon-Thurston map does not exist. In [Forum Math. Sigma 1, Article ID e3 (2013; Zbl 1276.20054)], \textit{O. Baker} and \textit{T. R. Riley} construct the first example of such an inclusion where the Cannon-Thurston map does not exist.
Let \(F_N\) be the free group of rank \(N\geq 2\) and \(\Phi\in\Aut(F_N)\), then the mapping torus group of \(\Phi\) is \(G_\Phi=F_N\rtimes_\Phi\langle t\rangle\). Since the inclusion \(F_N\leq G_\Phi\) depends only on the outer automorphism class \(\varphi\in\text{Out}(F_N)\) of \(\Phi\), we have the short exact sequence \(1\to F_N\to G_\varphi\to\langle t\rangle\to 1\). If the group \(G_\varphi\) is word-hyperbolic (in this case the automorphism \(\Phi\) (or \(\varphi\)) is called hyperbolic), then by the above mentioned Mitra's result, there does exist a continuous \(F_N\)-equivariant surjective Cannon-Thurston map \(\widehat\iota\colon\partial F_N\to\partial G_\varphi\).
In the present paper the authors study this Cannon-Thurston map \(\widehat\iota\). Before stating their main results we quote some definitions and terminology referring for details to the paper.
An automorphism \(\Phi\in\Aut(F_N)\) or its associated outer automorphism \(\varphi\in\text{Out}(F_N)\) is called fully irreducible or `iwip' if there is no non-trivial proper free factor of \(F_N\) which is mapped by any positive power of \(\Phi\) to a conjugate of itself. -- An automorphism \(\Phi\in\Aut(F_N)\) or its associated outer automorphism \(\varphi\in\text{Out}(F_N)\) is called atoroidal if no positive power of \(\Phi\) fixes any non-trivial conjugacy class \([w]\subseteq F_N\).
For any iwip automorphism \(\varphi\in\text{Out}(F_N)\) the following are equivalent.
(1) The automorphism \(\varphi\) is atoroidal.
(2) The automorphism \(\varphi\) is not induced by a homeomorphism of a surface with boundary.
(3) The mapping torus group \(G_\varphi\) is word-hyperbolic.
A point \(S\in\partial G_\varphi\) is called rational if it is the fixed point of an element \(g\in G_\varphi\setminus\{1\}\). If \(S=\lim_{n\to\infty}g^n\) (in the topology of the Gromov compactification of hyperbolic groups), then we write \(S=g^\infty\).
Let \(S\in\partial G_\varphi\). The degree \(\deg(S)\) of \(S\) denotes the cardinality of the full preimage of \(S\) under the map \(\widehat\iota\colon\partial F_N\to\partial G_\varphi\).
The following classes of points \(S\in\partial G_\varphi\) are defined: The point \(S\) is simple if \(\deg(S)=1\). The point \(S\) is regular if \(\deg(S)=2\). The point \(S\) is singular if \(\deg(S)\geq 3\). -- The regular and singular points are subdivided into two types. The point \(S\) is of \(\varphi\)-type if for every two distinct \(\widehat\iota\)-preimages \(X,Y\in\partial F_N\) of \(S\), \((X,Y)\in L(T_-)\). The point \(S\) is of \(\varphi^{-1}\)-type if for every two distinct \(\widehat\iota\)-preimages \(X,Y\in\partial F_N\) of \(S\), \((X,Y)\in L(T_+)\); where \(L(T_-)\) and \(L(T_+)\) are laminations defined in the paper.
Theorem 1. Let \(\varphi\in\text{Out}(F_N)\) be an atoroidal iwip and let \(\widehat\iota\colon\partial F_N\to\partial G_\varphi\) be the Cannon-Thurston map. Then one has \(\sum(\deg([S]_{F_N})-2)\leq 2N-2\), where the summation is taken over all \(F_N\)-orbits \([S]_{F_N}\) of singular points \(S\in\partial G_\varphi\) that are of \(\varphi\)-type.
The same inequality holds if the summation is taken over all \(F_N\)-orbits \([S]_{F_N}\) of singular points of \(\varphi^{-1}\)-type.
Theorem 2. Let \(\varphi\in\text{Out}(F_N)\) be an atoroidal iwip and let \(\widehat\iota\colon\partial F_N\to\partial G_\varphi\) be the Cannon-Thurston map. Then the following hold.
(1) For every \(S\in\partial G_\varphi\), we have \(\deg(S)\leq 2N\).
(2) The number of \(F_N\)-orbits of singular points of \(\varphi\)-type (respectively, of \(\varphi^{-1}\)-type ) in \(\partial G_\varphi\) satisfies \(\text{card}\{F_N\cdot S\subseteq\partial G_\varphi\mid S\) singular of \(\varphi\)-type \(\}\leq 2N-2\).
(3) Every singular point \(S\in\partial G_\varphi\) is rational. More precisely, there exists \(g\in G_\varphi\setminus F_N\) such that \(S=g^\infty\).
Theorem 3. Let \(\varphi\in\text{Out}(F_N)\) be an atoroidal iwip, let \(\widehat\iota\colon\partial F_N\to\partial G_\varphi\) be the Cannon-Thurston map and let \(g\in G_\varphi\setminus\{1\}\) be arbitrary. Then \(\deg(g^\infty)+\deg(g^{-\infty})\leq 4N-1\).
The upper bounds given in the theorems above are sharp. (For a concrete example, for every \(N\geq 3\) the authors refer to [\textit{A. J??ger} and \textit{M. Lustig}, Geom. Topol. Monogr. 14, 321-333 (2008; Zbl 1140.20027)].)
The paper concludes with the use of a proposition (Proposition 4.5 in the paper) to fill a gap in the proof of a Theorem of \textit{M. Mitra} [in Proc. Am. Math. Soc. 127, No. 6, 1625-1631 (1999; Zbl 0918.20028)] (a correction obtained already by Mitra himself [in ``On a theorem of Scott and Swarup'', \url{arXiv:1209.4165}]).
Dimitrios Varsos (Ath??na)
Zbl 1136.57009; Zbl 0880.57001; Zbl 0907.20038; Zbl 0914.20034; Zbl 1276.20054; Zbl 1140.20027; Zbl 0918.20028