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On property (T) for \(\operatorname{Aut}(F_n)\) and \(\mathrm{SL}_n(\mathbb{Z})\). (English) Zbl 1483.22006

In the 1960s, D. A. Kazhdan [Funct. Anal. Appl. 1, 63–65 (1967; Zbl 0168.27602); translation from Funkts. Anal. Prilozh. 1, No. 1, 71–74 (1967)] introduced the original definition of Property (T). It was stated in a representation-theoretic way. Kazhdan showed that a locally compact group with property (T) is compactly generated. Moreover, he showed that a lattice \(\Gamma\) in a locally compact group \(G\) has Property (T) if and only if so does \(G\).
Now, it is known that there are several equivalent conditions for groups to have Property (T). So far, Property (T) has actively been studied by a large number of authors, and has made brilliant progress. Today, the study of Property (T) includes a diverse range of research fields in mathematics, for example group theory, representation theory, differential geometry, the theory of group cohomology, geometric group theory, graph theory, ergodic theory and so on. For motivated readers, there is a remarkable detailed textbook by B. Bekka et al. [Kazhdan’s property. Cambridge: Cambridge University Press (2008; Zbl 1146.22009)].
Let \(F_n\) be the free group of rank \(n\), and \(\operatorname{Aut} F_n\) the automorphism group of \(F_n\). In this landmark paper, the authors showed that \(\operatorname{Aut} F_n\) has Property (T) for \(n \geq 6\).
Historically, the automorphism groups of free groups were begun to study by Dehn and Nielsen in the 1910s from a viewpoint of the low dimensional topology. In particular, Nielsen gave the first finite presentations for it. Over the last one century, multiple facets of the automorphism groups of free groups have been studied by a large number of authors, being compared with important groups including the mapping class groups of surfaces, the braid groups, the general linear groups over the integers and so on.
For the special linear groups over the integers, it is well-known that \(\mathrm{SL}(n,\mathbb Z)\) has Property (T) for \(n \geq 3\) due to Kazhdan since \(\mathrm{SL}(n,\mathbb Z)\) is a lattice in \(\mathrm{SL}(n,\mathbb R)\) having Property (T) for \(n \geq 3\). On the other hand, this fact was also shown directly by Y. Shalom [Publ. Math., Inst. Hautes Étud. Sci. 90, 145–168 (1999; Zbl 0980.22017)] who gave an explicit Kazhdan constant for \(\mathrm{SL}(n,\mathbb Z)\) by using a notion of bounded generation.
The group \(\operatorname{Aut} F_n\) is often compared with the general linear group \(\mathrm{GL}(n,\mathbb Z)\) through the natural surjection \(\rho : \operatorname{Aut} F_n \rightarrow \mathrm{GL}(n,\mathbb Z)\) induced from the abelianization of \(F_n\). The group \(\operatorname{Aut} F_2\) does not have Property (T) since \(\operatorname{Aut} F_2\) surjects onto \(\mathrm{GL}(2,\mathbb Z)\) which does not have Property (T).
For \(n=3\), the fact that \(\operatorname{Aut} F_3\) does not have Property (T) is obtained from independent works of J. McCool [Math. Proc. Camb. Philos. Soc. 106, No. 2, 207–213 (1989; Zbl 0733.20031)], and F. Grunewald and A. Lubotzky [Geom. Funct. Anal. 18, No. 5, 1564–1608 (2009; Zbl 1175.20028)]. For \(n=4\), the problem is still open. M. Kaluba et al. [Math. Ann. 375, No. 3–4, 1169–1191 (2019; Zbl 1494.22004)] showed that \(\operatorname{Aut} F_5\) has Property (T). Combining with these former results and the main result of the paper, we see that \(\operatorname{Aut} F_n\) has Property (T) for \(n\ge 5\).
In this paper, the authors adopt the following definition of Property (T) due to N. Ozawa [J. Inst. Math. Jussieu 15, No. 1, 85–90 (2016; Zbl 1336.22008)]. Let \(G\) be a group with a finite symmetric generating set \(S\). In the real group algebra \(\mathbb R[G]\) of \(G\), the element \[ \Delta := |S|- \sum_{s \in S} s= \frac{1}{2} \sum_{s \in S} (1-s)^*(1-s) \] is called the Laplacian of \(G\) with respect to \(S\) where the map \(* : \mathbb R[G] \rightarrow \mathbb R[G]\) is induced by \(g \mapsto g^{-1}\) for any \(g \in G\). The group \(G\) is said to have Property (T) if there exist \(\lambda>0\) and finitely many elements \(\xi_i \in\mathbb R[G]\) such that \[ \Delta^2- \lambda \Delta=\sum_i \xi_i^* \xi_i. \]
Let \(\mathrm{SAut}\,F_n\) be the preimage of \(\mathrm{SL}(n,\mathbb Z)\) by \(\rho\). It is called the special automorphism group of \(F_n\), and is of index \(2\) in \(\operatorname{Aut} F_n\). It has a finite presentation whose generators are all Nielsen transvections due to S. M. Gersten [J. Pure Appl. Algebra 33, 269–279 (1984; Zbl 0542.20021)].
In this paper, for \(G=\mathrm{SAut}\,F_n\) and \(S\) being set of all Nielsen transvections, the authors give an explicit estimate on Kazhdan constants and show that the Kazhdan radius is at most \(2\). By using it, the authors prove that \(\mathrm{SAut}\,F_n\) has Property (T) for \(n \geq 6\).
As a corollary, it is seen that \(\operatorname{Aut} F_n\) and the outer automorphism group \(\mathrm{Out}\,F_n\) have Property (T) for \(n \geq 6\).
The authors’ technique can be applied to the case where \(G=\mathrm{SL}(n,\mathbb Z)\) and \(S\) is the set of all elementary matrices for \(n \geq 3\). This means that the authors give a new proof for the fact that \(\mathrm{SL}(n,\mathbb Z)\) has Property (T) for \(n \geq 3\).
This excellent work by the authors will hold a place in the page of history for the study of the automorphism groups of free groups.

MSC:

22D55 Kazhdan’s property (T), the Haagerup property, and generalizations
20F28 Automorphism groups of groups

Software:

SCS; JuMP; GAP; Julia; Hecke; Nemo
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References:

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