\({\text {Aut}}({\mathbb {F}}_5)\) has property (T)
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Abstract
We give a constructive, computerassisted proof that \({\text {Aut}}({\mathbb {F}}_5)\), the automorphism group of the free group on 5 generators, has Kazhdan’s property (T).
Kazhdan’s property (T) is a powerful rigidity property of groups with many applications. Few groups are known to satisfy the property and the main classes of examples are: lattices in higher rank Lie groups, whose (rigid) algebraic structure allows to deduce various properties, including property (T); groups acting on complexes whose links satisfy a certain spectral condition, including groups acting on buildings; and certain hyperbolic groups, such as lattices in \({\text {Sp}}(n,1)\) and some random hyperbolic groups in the Gromov density model.
The main purpose of this article is to prove property (T) for a new group and give an estimate of its Kazhdan constant. Let \({\text {SAut}}({\mathbb {F}}_{5})\) denote the special automorphism group of the free group on 5 generators, with S being its standard generating set consisting of transvections.
Theorem 1
This method of proving property (T) was forseen by the third author in [29] and used effectively by Netzer and Thom [26], Fujiwara and Kabaya [12] and the first two authors [18] to reprove property (T) and give new estimates for Kazhdan constants for the groups \({\text {SL}}_n({\mathbb {Z}})\), \(n=3,4,5\). We describe in detail the theoretical aspects, the algorithm which is used to produce the solution, as well as the certification procedure that allows to obtain, out of this approximate solution, a mathematically correct conclusion about the existence of an exact one. In particular we describe the modifications to the algorithm of [18] which made a search for \(\xi _i\)s in the context of \({\text {SAut}}({\mathbb {F}}_{5})\) and the certification of the result technically possible.
As an immediate consequence of Theorem 1 we obtain
Corollary 2
The groups \({\text {Aut}}({\mathbb {F}}_5)\) and \({\text {Out}}({\mathbb {F}}_5)\) have property (T).
Questions whether any of the groups above has property (T) is discussed in many places, e.g. [6, Question 7], [22, 5, p 345], [4, p 4], [9, p 63], to name a few. Applications to the product replacement algorithm were discussed in [22].
The group \({\text {Aut}}({\mathbb {F}}_n)\) is known not to have property (T) for \(n=2\) and \(n=3\) see [24] as well as [4, 15]. It was suspected nevertheless that \({\text {Aut}}({\mathbb {F}}_n)\) might have property (T) for n sufficiently large, with all \(n\geqslant 4\) being open. In the case of \(n=4\) our approach—both the one in [18] and the symmetrized version presented below – did not give a positive answer, in the sense that we were not able to obtain a sufficiently approximate solution on the ball of radius 2. There are three possible reasons for this behavior. One possibility is that \({\text {Aut}}({\mathbb {F}}_4)\) does not have property (T) (as somehow anticipated by [4]). Another is that \({\text {Aut}}({\mathbb {F}}_4)\) has property (T), however no \(\xi _i\) as above, supported on the ball of radius 2 exist. Search for \(\xi _i\)s supported on the ball of radius 3 is already too expensive (in terms of memory and computation time) to be handled by our implementation. Finally, the third possibility is that again \({\text {Aut}}({\mathbb {F}}_4)\) has property (T) which is wittnessed on the ball of radius 2, but the spectral gap is so small, that the certification process does not yield a positive answer.
It is an interesting question whether the fact that \({\text {Aut}}({\mathbb {F}}_5)\) has property (T) is sufficient to deduce property (T) for \({\text {Aut}}({\mathbb {F}}_n)\) for all \(n\geqslant 5\), similarly as in the case of lattices in higher rank Lie groups. In Sect. 6 we show that \({\text {Aut}}({\mathbb {F}}_{n+1})\) has a large subgroup with property (T), if \({\text {Aut}}({\mathbb {F}}_n)\) has (T). However, all attempts to prove property (T) for \({\text {Aut}}({\mathbb {F}}_n)\), \(n\geqslant 6\) using property (T) for \({\text {Aut}}({\mathbb {F}}_5)\) seem to break down at the currently open Question 12 in [6].
It was proved in [14] that for \(n\geqslant 3\) the group \({\text {Out}}({\mathbb {F}}_n)\) is residually finite alternating and it is a consequence of Theorem 1 that the corresponding family of alternating quotients of \({\text {Out}}({\mathbb {F}}_5)\) can be turned into a family of expanders. This was proved earlier in greater generality by Kassabov [20], however in our case the generating set is explicit and the same in all of these finite groups, in the sense that it is the image of the generating set of \({\text {Out}}({\mathbb {F}}_5)\). In particular, this gives an alternative and independent of [20] negative answer to question [21, Question 10.3.2].
Theorem 1 also allows us to give an answer to a question of Popa on the existence of certain crossed product von Neumann algebras with property (T), see Remark 12.
Note: The techniques developed here became later crucial in [17] while proving property (T) for \({\text {Aut}}(F_n)\) for all \(n\geqslant 6\). However, the case \(n=5\) is not accessible via the argument in [17] and the only existing proof of the case \(n=5\) is in the current paper.
1 Property (T), real algebraic geometry and semidefinite programming
Let G be a discrete group generated by a finite set \(S=S^{1}\). Given a ring R we consider the associated group ring RG, that consists of finitely supported functions \(\xi :G\rightarrow R\). We will use the notation \(\xi =\sum _{g\in G} \xi _g g\), where \(\xi _g\in R\) for each \(g\in G\), to denote the elements of the group ring RG. The product in RG is then defined by the convolution \((\xi \eta )_g=\sum _{h\in G} \xi _{h}\eta _{h^{1}g}\). Recall that the augmentation ideal IG is the kernel of the augmentation map \(\omega :RG\rightarrow R\), \(\xi \mapsto \sum _{g\in G} \xi _g\). The group ring RG is equipped with an involution \(^*:RG\rightarrow RG\), induced by the inversion map on G and defined explicitly as \((\xi ^*)_g=\xi _{g^{1}}\) for any \(\xi \in R G\).
Theorem 3
The characterization above can be used to prove property (T) for a particular group G by providing an explicit solution of Eq. (1). In particular, the fact that the right hand side is a finite sum of (hermitian) squares of finitely supported functions allows us to try to obtain the \(\xi _i\)’s using semidefinite programming.
The process consists of finding (the real part of) the square root Q of P (i.e. \(QQ^T \approx P\)), and projecting the obtained matrix Q onto the augmentation ideal, simply by subtracting the mean value of the ith column of Q from each entry in that column. This procedure, e.g. performed in rational (multiprecision) arithmetic, provides an explicit matrix \({\overline{Q}}\), whose columns correspond to elements of the augmentation ideal supported on E. In our case this is done in interval arithmetic as described in detail later. This process introduces additional error into each element of \({\mathbf {x}}^*{\overline{Q}}\left( {\mathbf {x}}{\overline{Q}}\right) ^T = \sum _i \xi _i^*\xi _i\), which in general depends on the accuracy obtained by the chosen numerical solver. However (and most importantly) after the projection, Lemma 4 below allows to dominate \(r = \Delta ^2  \lambda _0\Delta  \sum _i \xi _i^*\xi _i\), the remainder of the solution. Note that r gathers both the inaccuracy of the solver and the error introduced by the projection.
For \(\xi \in {\mathbb {R}}G\) let \(\Vert \xi \Vert _1=\sum _{g\in G}\vert \xi _g\vert \) be the norm of \(\xi \) in \(\ell _1(G)\). We write \(a \geqslant b\) for elements a, b of a group ring to denote that \(ab\) enjoys a decomposition into sum of (hermitian) squares. The following lemma allows to estimate the magnitude of errors involved in our computations and makes certification possible. For a more thorough treatment of order units see [29], as well as [16, 31] for a more general context.
Lemma 4
1.1 Complexity of the problem
The size of the set E translates directly to the computational complexity of optimization problem (OP): while each element \(\xi _i\) is (by its definition) supported on E, Eq. (1) defines \(E^{1}E\) linear constraints and \(E^2\) variables in one semidefinite constraint of size \(E\times E\). It seems to be an interesting problem to understand the influence of the choice of the set E on the obtained bound \(\lambda _0\) and numerical properties of the problem.
The optimization problem has been solved numerically in several cases in [12, 18, 26] yielding new estimates for \(\lambda \). The computations were successful for \(E=B_2(e,S)\) e.g. for the groups \({\text {SL}}_n({\mathbb {Z}})\) with \(n=3,4,5\). Finding a numerical solution of the problem on a computer may not be feasible if the size of the set E is too large, and in fact this is the case for the groups \({\text {SL}}_n({\mathbb {Z}})\), \(n\geqslant 6\), or the groups \({\text {SAut}}({\mathbb {F}}_{n})\) when \(n\geqslant 4\). For instance, the case of \({\text {SAut}}({\mathbb {F}}_{5})\) results in 21, 538, 881 variables and 11, 154, 301 constraints, making it a prohibitively large problem. In order to remedy this and to be able to find a solution to optimization problem (OP) e.g. for \({\text {SAut}}({\mathbb {F}}_{5})\), we will use the symmetries of the set E to reduce the problem’s size.
2 Problem symmetrization
The size and computational complexity of optimization problem (OP) can be significantly decreased by exploiting its rich symmetry derived from the group structure. Roughly speaking, we will replace solving a large problem by solving many smaller problems and patching the solutions together to obtain a solution to the original, larger problem. While there are \(E^2\) variables in the original problem, the number of variables of the symmetrized version is \(m_1^2+\cdots +m_k^2\), where \(m_k\) are the dimensions of the individual component problems. Note that \(\sum _i m_i \leqslant E\) i.e. the latter is much smaller than the former. Moreover using orbit constraints we will reduce the number of constraints significantly. In the case of \({\text {SAut}}({\mathbb {F}}_{5})\) the result is a problem consisting of 13,232 variables in 36 semidefinite constraints and 7229 linear constraints, which can be realistically attacked with a numerical solver. However, to be able to perform our computations we will need to give a different parametrisation of the space of possible solutions by the means of orbit reduction and the Wedderburn decomposition.
A somewhat parallel exposition of semidefinite programs size reduction using its symmetry is discussed in detail in [1, 7]. We would like to point out that a numerical approach to numerical symmetrization of optimization problems that does not use group representation theory is described in [25], which may be applicable e.g. to finitely presented groups where the symmetry is not clearly visible.
2.1 Invariant SDP problems
Lemma 5
The expression \(\Delta ^2\lambda \Delta \) is invariant under \(\Sigma \).
Proof
Lemma 6
We have \(\delta _{\sigma (t)}=\sigma (\delta _t)\) and \(\delta _{t^{1}}=\delta _t^T\). \(\square \)
In [1] a semidefinite problem is said to be invariant with respect to an action of a group G if for every solution P, gP is also a solution for every \(g\in G\).
Proposition 7
Optimization problem (OP) is \(\Sigma \)invariant.
Proof
In particular, convexity yields
Corollary 8
Let \(P\in {\mathbb {M}}_E\) be a solution to problem (OP) for some \(\lambda >0\). Then there exists \(P\in {\mathbb {M}}_E^{\Sigma }\) that also solves (OP) for the same \(\lambda >0\).
The corollary above shows that we may as well search for an invariant solution.
2.2 Orbit symmetrization
2.3 Blockdiagonalization via Wedderburn decomposition
Recall that the orthogonal dual \({\widehat{G}}\) of a group G is the family of equivalence classes of irreducible orthogonal representations of G. We will work under the assumption that all irreducible characters of \(\Sigma \), the subgroup of the group of symmetries of E, are real. By \(n{\mathbf {1}}_{\mathbb {R}}\) we denote the nfold direct sum of the trivial, 1dimensional real representation.
It follows from the definition of \(p_\pi \) that the image of \(\varrho _E(p_\pi )\) is a subspace of \(\ell _2(E)\) of dimension \(m_\pi \).
2.4 Symmetrization
For the purposes of numerical computation, instead of working with \(\Theta \) and a basis of \({\mathbb {M}}_E^\Sigma \) it is advantageous to work with the standard bases of each of \({\mathbb {M}}_{m_\pi }\) and translate the constraints to the new basis via \(\Theta \).
Proposition 9
Proof
3 The group \({\text {Aut}}({\mathbb {F}}_n)\)
3.1 Presentation for \({\text {Aut}}({\mathbb {F}}_n)\)
3.2 Implementation of \({\text {Aut}}({\mathbb {F}}_n)\)
When computing in \({\text {Aut}}({\mathbb {F}}_{n})\) one usually choses to represent elements either as words in a finite presentation, or actual functions on \({\mathbb {F}}_n\) transforming free generating sets of \({\mathbb {F}}_n\) to free generating sets. The former aproach allows the group operations to be purely mechanical operations on symbols, but the recognition of the identity element (the word problem) is a major obstruction to effective computation. The latter approach, requires storage of both the domain (a generating ntuple) and the image of the domain. It provides an easier solution to the recognition of the identity problem: two automorphisms are equal if and only if their values on the standard generating set of \({\mathbb {F}}_n\) agree. However, to compute those values one needs to “equalize” the domains and compare the images of the domain under both automorphims. The final step requires only trivial cancellation.
Instead of using either of the approaches we decided to produce our own implementation, where each element carries both the structure of a word in a finite presentation, as well as functional information. Let \(\Gamma _n\) be the graph with the vertex set consisting of generating ntuples of \({\mathbb {F}}_n\) and an edge connecting two such tuples if one can be obtained from the other by the application of an automorphism \(s\in S\) (we label the edge by s). We represent elements of \({\text {Aut}}({\mathbb {F}}_n)\) as paths in the graph \(\Gamma _n\) which start at the standard ngenerating tuple \((x_1 , \ldots , x_n )\). Such path is represented by a word over alphabet S by collecting edge labels in a natural fashion. Therefore each such path determines an automorphism of \({\mathbb {F}}_n\) which takes the tuple \((x_1 , \ldots , x_n )\) (the initial vertex of the path) to the ngenerating tuple of the terminal vertex.
Effectively, we represent \({\text {Aut}}({\mathbb {F}}_n)\) as a finitely presented group on S and solve the word problem in an indirect manner. Note that in this setting it is sufficient to store in each letter only minimal amount of information (namely: its type (R, or L), two indicies (i, j) and the exponent (±)), as storage of neither the reference basis nor the image is required.
3.3 The symmetrizing group \(\Sigma \)
The group generated by the automorphisms permuting and inverting generators of \({\mathbb {F}}_n\) is isomorphic to the group \({\mathbb {Z}}/_2\wr S_n\) (the signed permutation group). Consider the action of \({\mathbb {Z}}/_2\wr S_n\) on \({\text {Aut}}({\mathbb {F}}_n)\) by conjugation. Clearly the action preserves the set S of all Nielsen transformations and hence the group \({\text {SAut}}({\mathbb {F}}_{n})\). Moreover the action preserves the wordlength metric on \({\text {SAut}}({\mathbb {F}}_{n})\) induced by S (and thus \(E=B_2(e,S)\)), so we can set \(\Sigma = {\mathbb {Z}}/_2\wr S_n\) in our considerations.
3.3.1 Minimal projection system for \(\Sigma \)
3.3.2 Minimal projection system for \(S_n\), \(n\leqslant 6\)
n  Partition  \(\varepsilon _\pi \) 

3  \(2_1 1_1\)  \(\frac{1}{2}\big (() + (1,2)\big )\) 
4  \(2_1 1_2\)  \(\frac{1}{2}\big (() + (1,2)\big )\) 
\(3_1 1_1\)  \(\frac{1}{2}\big (()  (1,2)\big )\)  
\(2_2 \)  \(\frac{1}{2}\big (() + (1,2)\big )\)  
5  \(2_1 1_3 \)  \(\frac{1}{2}\big (() + \text {(}1,2\text {)}\big )\) 
\(3_1 1_2 \)  \(\frac{1}{4}\big (() + (1,4,3,2) + (1,3)(2,4) + (1,2,3,4)\big )\)  
\(2_2 1_1 \)  \(\frac{1}{3}\big (() + (1,3,2) + (1,2,3)\big )\)  
\(4_1 1_1 \)  \(\frac{1}{2}\big (()  \text {(}1,2\text {)}\big )\)  
\(3_1 2_1 \)  \(\frac{1}{3}\big (() + (1,3,2) + (1,2,3)\big )\)  
6  \(2_1 1_4 \)  \(\frac{1}{2} \big (() + (1,2)\big )\) 
\(3_1 1_3 \)  \(\frac{1}{4} \big (() + (1,2) + (1,2)(3,4) + (3,4)\big )\)  
\(2_2 1_2 \)  \(\frac{1}{5} \big (() + (1,3,5,2,4) + (1,4,2,5,3) + (1,5,4,3,2) + (1,2,3,4,5)\big )\)  
\(4_1 1_2 \)  \(\frac{1}{4} \big (()  (1,2) + (1,2)(3,4)  (3,4)\big )\)  
\(3_1 2_1 1_1 \)  \(\frac{1}{12} \big (() + (1,2) + (1,2)(3,4) + (3,4) + (3,4,5) + (1,2)(3,4,5) + \)  
\((1,2)(3,5) + (3,5) + (1,2)(4,5) + (4,5) + (3,5,4) + (1,2)(3,5,4)\big )\)  
\(5_1 1_1 \)  \(\frac{1}{2} \big (()  (1,2)\big )\)  
\(2_3 \)  \(\frac{1}{3} \big (() + (1,3,2) + (1,2,3)\big )\)  
\(4_1 2_1 \)  \(\frac{1}{5} \big (() + (1,3,5,2,4) + (1,4,2,5,3) + (1,5,4,3,2) + (1,2,3,4,5)\big )\)  
\(3_2 \)  \(\frac{1}{3} \big (() + (1,3,2) + (1,2,3)\big )\) 
4 Description of the algorithm
 1.
generate \(E=B_2(e,S)\), \(E^{1}E = B_4(e,S)\) and \(\Delta \) (stored in delta.jld as the coefficients vector in \(E^{1}E\));
 2.
compute the division table \(E^{1}\times E \rightarrow E^{1}E\) (stored in pm.jld);
 3.
compute the permutation representation \(\varrho _E:\Sigma \rightarrow {\mathbb {M}}_E\) (stored in preps.jld);
 4.
compute the Wedderburn decomposition of \({\mathbb {M}}_E\), i.e. for every \(\pi \in {\widehat{\Sigma }}\) compute \(U_\pi \)’s (stored in U_pis.jld)
 5.
decompose \(E^{1}E\) into orbits of \(\Sigma \) (stored in orbits.jld);
 6.
use \(\Delta \), the orbit structure, the division table and \(U_\pi \)’s to construct the constraints of symmetrized optimization problem (SOP);
 7.
solve the symmetrized optimization problem to obtain \(\big (\lambda _0, \{P_\pi \}_\pi \big )\) (stored in lambda.jld, and SDPmatrix.jld);
 8.
reconstruct P according to Proposition 9 (stored in SDPmatrix.jld);
 9.
certify the solution \((\lambda _0, P)\) as described in Sect. 4.3.
4.1 Division table on E
To perform quickly the multiplication \(\xi _i^*\xi _i\) we cache the division table \(M:E^{1}\times E \rightarrow E^{1}E\), as a matrix \(M\in {\mathbb {M}}_E\) such that \(M_{g,h} = g^{1}h\). To avoid indexing entries of M by group elements and storing them in M (due to technical reasons) we fix a nondecreasing (with the wordlength) order \({\mathbf {x}}\) of elements in \(E^{1}E\), i.e. such that \({\mathbf {x}}_0 = e\), \(\{{\mathbf {x}}_j :j \in \{1, \ldots , S\}\}=S\), and \(E \subseteq E^{1}E\) as the first Eelements. Then we can store only the (integer) indices of elements in the division table, i.e. if \(g \in E\) is the ith element of \(E^{1}E\), (\(g = {\mathbf {x}}_i\)) and \(h\in E\) is the jth element of \(E^{1}E\) (\(h={\mathbf {x}}_j\)), then \(M_{i,j} = k\), where \(g^{1}h = {\mathbf {x}}_k\) is the kth element of \(E^{1}E\). Thus once the full M has been populated, we no longer need the actual group elements to perform (twisted^{1}) multiplication of elements of \({\mathbb {R}}G\) supported on E. In particular, given a solution \((\lambda _0, P)\) of problem (OP), division table M and \(\Delta \) (as vector of values on \({\mathbf {x}}\)), one can compute the sum of squares decomposition \(\sum _i \xi _i^*\xi _i\) and compare it with \(\Delta ^2  \lambda _0 \Delta \) without the need to access group elements directly.
Note that the full division table is also needed for producing the constraint matrices.
4.2 Symmetrization
The minimal projection system \(\{p_\pi \}_{\pi \in {\widehat{\Sigma }}}\) for \(\Sigma \) is computed in \({\mathbb {Q}}\Sigma \) as described in Sect. 3.3. Then coefficients are converted to floating point numbers and \(\varrho _E(p_\pi )\) is evaluated, where \(\varrho _E\) is the permutation representation of \(\Sigma \) on E. Matrix representatives for \(U_\pi \) are obtained from \(\varrho _E(p_\pi )\) using singular value decomposition.
4.3 Certification
4.4 Software details

package Groups.jl for computations in wreath products and automorphism groups of free groups;

package GroupRings.jl for computations in group rings (with basis);

package PropertyT.jl for computations of the spectral gap;
4.5 Replication details
The ball of radius 4 in \({\text {SAut}}({\mathbb {F}}_{5})\) consists of 11,154,301 elements. Generating \(B_4(e, S)\), decomposing it into 7229 orbits of \({\mathbb {Z}}_2\wr S_5\) action, computing division table on \(B_2(e, S)\) and finding \(U_\pi \)’s for the Wedderburn decomposition takes about 3 h and requires at most \(20\hbox {GB}\) of RAM.^{2} Once this has been computed, the actual optimization problem consists of 13,232 variables in 36 semidefinite blocks and 7230 constraints. The optimization phase had been running for over 800 h until the acuracy of \(10^{12}\) has been reached. Note that a much shorter running time is possible if one additionally constraints \(\lambda \) from above, by say 1.0. The reconstruction of P (according to Proposition 9) and its certification take approximately 1.5 h in total. The replication of the results should be possible on a reasonably modern desktop computer (times reported correspond to a workstation with 4core CPU).
The precomputed division table, orbit decomposition, \(U_\pi \)’s, as well as the solution P used in the proof of Theorem 1 can be obtained from [19].
5 Proof of Theorem 1
We are now in position to prove our main theorem. As indicated above we set \(E = B_2(e,S)\) and obtained a solution of optimization problem (SOP).
Proof of Theorem 1
6 Extrapolation of property (T)
In case of arithmetic groups, once we know \({\text {SL}}_n({\mathbb {Z}})\) has property (T) for some n, it is rather easy to deduce from this fact that \({\text {SL}}_m({\mathbb {Z}})\) has property (T) for all \(m\geqslant n\), because \({\text {SL}}_m({\mathbb {Z}})\) is boundedly generated by finitely many conjugates of \({\text {SL}}_n({\mathbb {Z}})\) (see [33], particularly around Section 4.III.7).
However, in the case of \({\text {Aut}}({\mathbb {F}}_n)\) a similar approach seems to break down at the currently open Question 12 in [6]. Namely, it is not known whether a quotient Q of \({\text {Aut}}({\mathbb {F}}_{n+1})\) must be finite provided that \({\text {Aut}}({\mathbb {F}}_n)\) has finite image in Q. If a counterexample exists, property (T) of \({\text {Aut}}({\mathbb {F}}_n)\) cannot, obviously, tell anything about such an infinite quotient Q. We nevertheless make an effort to extrapolate property (T) of \({\text {Aut}}({\mathbb {F}}_n)\) to a larger group.
Proposition 10
Proof
The above proposition applies to the action of \({\text {Aut}}({\mathbb {F}}_n)\) on \({\mathbb {F}}_n=\langle x_1,\ldots ,x_n\rangle \) for \(n\geqslant 2\), since \(x_i^m = x_j^{1}\cdot x_j^{} x_i^m\)\((j\ne i)\) has \(\ell (x_i^m )\leqslant 2\).
Corollary 11
Proof
Let \(\left\{ x_i\right\} _{i=1}^{n+1}\) denote the standard generating set of \({\mathbb {F}}_{n+1}\). Any element \(\theta \) as above satisfies \(\theta (x_{n+1})=a x_{n+1}^{\pm 1} b\) for some \(a,b\in {\mathbb {F}}_n\): indeed, since \(\theta ({\mathbb {F}}_{n}) \subseteq {\mathbb {F}}_n\), \(\theta \) has a word representative which involves neither letters \(L_{i,n+1}^{\pm 1}\) nor \(R_{i, n+1}^{\pm 1}\) for \(1\leqslant i \leqslant n\). Therefore any letter \(L_{n+1, i}^{\pm 1}\) (\(R_{n+1, i}^{\pm 1}\) respectively) of \(\theta \) results in multiplying \(x_{n+1}\) by a word from \({\mathbb {F}}_n\) on the left (right respectively). This means that \(\Gamma \) is isomorphic to \(({\text {Aut}}({\mathbb {F}}_n)\times {\mathbb {Z}}/2)\ltimes ({\mathbb {F}}_n\times {\mathbb {F}}_n)\), where \({\text {Aut}}({\mathbb {F}}_n)\) acts on \({\mathbb {F}}_n\times {\mathbb {F}}_n\) diagonally and \({\mathbb {Z}}/2\) acts on \({\mathbb {F}}_n\times {\mathbb {F}}_n\) by the flip. Since \({\text {Aut}}({\mathbb {F}}_n)\ltimes {\mathbb {F}}_n\) has property (T) by the above proposition, \({\text {Aut}}({\mathbb {F}}_n)\ltimes ({\mathbb {F}}_n\times {\mathbb {F}}_n)\) has property (T) as well. \(\square \)
Remark 12
In [30, 4) p. 324] Popa asked for an example of an action of a property (T) group G on \(L({\mathbb {F}}_n)\), the free group factor, so that the crossed product \(L({\mathbb {F}}_n)\rtimes G\) has property (T). The examples considered above provide an answer to Popa’s question for \(n=5\). Indeed, by the above arguments, taking \(G={\text {Aut}}({\mathbb {F}}_{5})\) with its natural action on \(L({\mathbb {F}}_5)\) satisfies all the necessary conditions. Another example is given by taking \(G={\text {Out}}({\mathbb {F}}_5)\), in which case \(L({\text {Aut}}({\mathbb {F}}_5))\) is the crossed product \(L({\mathbb {F}}_5)\rtimes {\text {Out}}(F_5)\).
7 \({\text {SL}}_6({\mathbb {Z}})\) and \({\text {SL}}_4({\mathbb {Z}}\langle X\rangle )\)
8 Final remarks
Smallest radius for existence of a solution As mentioned in the introduction, one of the reasons for our approach not producing an answer in the case of \({\text {Aut}}({\mathbb {F}}_4)\) could be that the Eq. (1) does not have a solution where all \(\xi _i\)’s are supported in ball of radius 2. It is in general unclear, for a group with property (T), what is the smallest radius r such that (1) has a solution on the ball of radius r. It is equally unclear what is the radius r for which the optimal \(\lambda \) is attained (if it exists).
We have not been able to reprove property (T), using the method presented here, for \({\text {SL}}_3({\mathbb {Z}}[X])\) on the ball of radius 2. Recall that for \({\text {SL}}_3({\mathbb {Z}}[X])\) property (T) was proved by Shalom [33] and Vaserstein [35], as well as by Ershov and JaikinZapirain [10].
Note also that the bound given in [12] on a ball of radius 2 was better for \({\text {SL}}_4({\mathbb {Z}})\) than for \({\text {SL}}_3({\mathbb {Z}})\). All this suggests that to detect property (T) (or estimate the Kazhdan constant) of \({\text {SL}}(n,{\mathbb {Z}}[X_1,\ldots ,X_k])\) one needs larger ball as k increases and smaller ball as n increases (although the Kazhdan constant itself gets smaller).
Footnotes
 1.
Note that given two vectors of values on \({\mathbf {x}}\) which represent elements x, y of RG, using only the division table (i.e. without referring to the elements of \({\mathbf {x}}\)) we can compute \(x^*y\), but not xy.
 2.
The initial computations have been performed on a cluster in PLGrid network.
Notes
Acknowledgements
MK is partially supported by the National Science Center, Poland Grant 2015/19/B/ST1/01458 and 2017/26/D/ST1/00103. NO is partially supported by JSPS KAKENHI, Grant number 17K05277 and 15H05739. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement no. 677120INDEX). This research was supported in part by PLGrid Infrastructure, Grant ID: propertyt2.
References
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