Ershov and Jaikin-Zapirain's spectral criteria for Property (T)

In a previous post we discussed Property (T) and some of the consequences of having (T). However, we would like a way to show lots of groups have (T), otherwise we can't unlock its consequences. What follows is a discussion of the landmark work of Ershov and Jaikin-Zapirain that gives a new spectral criterion for groups to have Property (T). They use this to show that the group of matrices generated by elementary matrices has (T), which has been of considerable interest. 

Definition Let \( G \) be a discrete group and \( B \) a subset of \( G \). The pair \( (G,B) \) has \emph{relative property (T)} if for any \( \varepsilon > 0 \) there are a finite subset \( S \subset G \) and \( \mu > 0 \) such that if \( V \) is any unitary representation of \( G \) and \( v \in V \) is \( (S,\mu) \)-invariant, then \( v \) is \( (B,\varepsilon) \)-invariant.

Remark The pair \( (G,B) \) may have relative property (T) even if \( G \) is not finitely generated: for instance, if \( B \) is a group with property (T), then \( (G,B) \) has relative property (T) for any overgroup \( G \). However, if \( G \) is generated by a finite set \( S_0 \), then in the definition of relative (T) for \( (G,B) \), we can require that \( S = S_0 \).

An important special case of relative property (T) is when the dependence of \( \mu \) on \( \varepsilon \) in the above definition may be expressed by a linear function. We reflect this property in the following definition.

Definition Let \( G \) be a discrete group and \( B \), \( S \) subsets of \( G \). The Kazhdan ratio \( \kappa_r(G,B;S) \) is the largest \( \delta \in \mathbb{R} \) with the following property: if \( V \) is any unitary representation of \( G \) and \( v \in V \) is \( (S,\delta \varepsilon) \)-invariant, then \( v \) is \( (B,\varepsilon) \)-invariant.

Clearly, if \( \kappa_r(G,B;S) > 0 \) for some finite set \( S \), then \( (G,B) \) has relative (T). On the other hand, if \( B \) is a normal subgroup of \( G \), then

\[\kappa_r(G,B;S) \geq \frac{\kappa(G,B,S)}{2]\]

(this inequality is essentially established in Corollary 2.3 of Shalom's paper). Thus, if \( B \) is a normal subgroup of \( G \), then relative property (T) for \( (G,B) \) is equivalent to the positivity of the Kazhdan ratio \( \kappa_r(G,B;S) \) for some finite set \( S \).

It is clear from definitions that if a group \( G \) has a subset \( B \) such that \( \kappa(G,B) > 0 \) and \( (G,B) \) has relative property (T), then \( G \) has property (T). If, in addition, we know that \( \kappa_r(G,B;S) > 0 \) for some finite set \( S \), we can estimate the Kazhdan constant \( \kappa(G,S) \) using the following inequality:

\[\kappa(G,S) \geq \kappa(G,B) \cdot \kappa_r(G,B;S). \]

This argument was used in Shalom’s proof of property (T) for \( \mathrm{SL}_n(\mathbb{Z}) \) as follows: it is first shown that

\[\kappa(\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2, \mathbb{Z}^2; F) \geq \frac{1}{10}\]

for some natural generating set \( F \) of \( \mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2 \), and thus

\[\kappa_r(\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2, \mathbb{Z}^2; F) \geq \frac{1}{20}\]

(since \( \mathbb{Z}^2 \) is a normal subgroup of \( \mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2 \)). Using natural embeddings of \( \mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2 \) into \( \mathrm{SL}_n(\mathbb{Z}) \), one concludes that

\[\kappa_r(\mathrm{SL}_n(\mathbb{Z}), U; \Sigma) \geq \frac{1}{20},\]

where \( \Sigma \) is the set of elementary matrices with 1 off the diagonal and \( U \) is the set of all elementary matrices in \( \mathrm{SL}_n(\mathbb{Z}) \). On the other hand, since \( \mathrm{SL}_n(\mathbb{Z}) \) is boundedly generated by elementary matrices, we have \( \kappa(\mathrm{SL}_n(\mathbb{Z}), U) > 0 \). Thus, by the inequality for the estimate on the Kazhdan constant, \( \mathrm{SL}_n(\mathbb{Z}) \) has property (T).

The strategy employed below will be similar, but showing the bounds on the relative Kazhdan constants becomes much more difficult, so a lot more analytic machinery is developed.

Hilbert space preliminaries

Definition Let \( V \) be a Hilbert space, and let \( X \) and \( Y \) be subspaces of \( V \). Let \( \varepsilon \geq 0 \) be a real number.

• We will say that \( X \) and \( Y \) are \emph{\( \varepsilon \)-orthogonal} and write \( X \perp_\varepsilon Y \) if \[  |\langle x, y \rangle| \leq \varepsilon \|x\| \, \|y\|\]  for any \( x \in X \) and \( y \in Y \).
•  We will say that \( X \) is \emph{\( \varepsilon \)-close} to \( Y \) if  \[  \mathrm{dist}(x, Y) \leq \varepsilon \|x\|  \] for any \( x \in X \), that is, for any \( x \in X \) there exists \( y \in Y \) such that \( \|x - y\| \leq \varepsilon \|x\| \).

We require a technical lemma on Hilbert space geometry in what follows:

Lemma 1 Let \( V \) be a Hilbert space, \( \{V_i\}_{i=1}^n \) subspaces of \( V \), and let \( \rho = \rho(\{V_i\}) \). Let \( x \in V \), and for each \( i \in \{1, \ldots, n\} \), write \( x = a_i + b_i \), with \( a_i \in V_i \) and \( b_i \in V_i^\perp \). Then there exists \( j \) such that 

\[\|b_j\| \geq \sqrt{1 - \rho} \|x\|.\]

Criteria for Property (T)

For a (topological) group $G$,by $\mathrm{Rep}(G)$ we will denote the class of (continuous) unitary representations of $G$, and by $\mathrm{Rep}_0(G)$ the class of (continuous) unitary representations of $G$ without nonzero invariant vectors.

Definition: Let \( G \) be a group.

•  Let \( H \) and \( K \) be subgroups of \( G \) such that \( G = \langle H, K \rangle \). We say that \( H \) and \( K \) are \( \varepsilon \)-orthogonal if for any \( V \in \mathrm{Rep}_0(G) \), the subspaces \( V^H \) and \( V^K \) are \( \varepsilon \)-orthogonal.
•     Let \( \{H_i\}_{i=1}^n \) be subgroups of \( G \). The codistance between \( \{H_i\} \) in \( G \), denoted \( \rho(\{H_i\}, G) \), is defined to be the supremum of the set  \[    \left\{ \rho(V^{H_1}, \ldots, V^{H_n}) : V \in \mathrm{Rep}_0(G) \right\}.\]

 If \( G = \langle H_1, \ldots, H_n \rangle \), we simply write \( \rho(\{H_i\}) \) instead of \( \rho(\{H_i\}, G) \). It is easy to see that if \( G = \langle H_1, \ldots, H_n \rangle \), then \( \rho(\{H_i\}, G) = 1 \).

Lemma 2 Let \( G \) be a group and \( H_1, \ldots, H_n \) subgroups of \( G \) such that \( G = \langle H_1, \ldots, H_n \rangle \). Let \( \rho = \rho(\{H_i\}) \), and suppose that \( \rho < 1 \). Then:

1.    \( \kappa(G, H_i) \geq \sqrt{2(1 - \rho)} \).
2.   Let \( S_i \) be a generating set of \( H_i \), and let \( \delta = \min\{\kappa(H_i, S_i)\}_{i=1}^n \). Then \[   \kappa(G, \cup S_i) \geq \delta \sqrt{1 - \rho}.\]
3. Assume in addition that each pair \( (G, H_i) \) has relative property (T). Then \( G \) has property (T).

Proof It is clear from the definitions that if each pair \( (G, H_i) \) has relative (T), then \( (G, H_i) \) also has relative (T). Hence (3) is a consequence of (1). By Proposition 1.1.5 of Bekka-de la Harpe-Valette, \( \kappa(H_i, H_i) \geq \sqrt{2} \) for any group \( H_i \), so (1) is a special case of (2) with \( S_i = H_i \). Thus, we only need to prove (2).

Let \( V \in \mathrm{Rep}_0(G) \), and take any nonzero \( v \in V \). For each \( i = 1, \ldots, n \), we write \( v = a_i + b_i \) where \( a_i \in V^{H_i} \) and \( b_i \in (V^{H_i})^\perp \). By Lemma 1, we have \( \|b_i\| \geq \|v\| \sqrt{1 - \rho} \) for some \( i \).

Note that \( (V^{H_i})^\perp \) is a unitary representation of \( H_i \) without invariant vectors. Since \( \kappa(H_i, S_i) \geq \delta \) and \( b_i \in (V^{H_i})^\perp \), there exists \( s \in S_i \) such that

\[\|s b_i - b_i\| \geq \delta \|b_i\| \geq \delta \|v\| \sqrt{1 - \rho}.\]

On the other hand,

\[\|s v - v\| = \|s a_i + s b_i - a_i - b_i\| = \|s b_i - b_i\|\]

since \( a_i \) is \( s \)-invariant. Thus, \( \kappa(G, S_i, V) \geq \delta \sqrt{1 - \rho} \). $\blacksquare$

In general, it seems very hard to establish property (T) for an infinite group \( G \) by a direct application of Lemma 2. However, in the paper two further results are proved which make it possible to prove property (T) for some complicated infinite groups, only applying Lemma 2 to much simpler groups whose representations are easily described.

Ershov-Jaikin-Zapirain manage to prove the results we will see about the Steinberg group using these ideas. However, they have been pushed further to get the following strengthening, due to Kassabov:

Theorem 3: Let \( G \) be a group generated by subgroups \( H_1, \ldots, H_n \) (where \( n \geq 2 \)), and let \( \varepsilon_{ij} = \varepsilon(H_i, H_j) \) for \( i \neq j \). Let \( E = (e_{ij}) \) be the \( n \times n \) matrix defined by \( e_{ii} = 1 \) and \( e_{ij} = -\varepsilon_{ij} \) for \( i \neq j \), and assume that \( E \) is positive definite. Then \( \kappa(G, H_i) > 0 \).

It is easy to see that the matrix \( E \) is positive definite in the following two special cases (proved originally in the Ershov-Jaikin-Zapirain paper):

1.  \( \max\{\varepsilon_{ij} : i \neq j\} < \frac{1}{n-1} \)
2.  \( n = 3 \) and \( \varepsilon_{12}^2 + \varepsilon_{23}^2 + \varepsilon_{13}^2 + 2 \varepsilon_{12} \varepsilon_{23} \varepsilon_{13} < 1 \)

The Steinberg group

We now apply this to show property (T) for the Steinberg group, which is of interest both to group theorists and to algebraic K-theorists.

Definition Let \( n \geq 3 \). Let $R$ be an associative ring. The Steinberg group \( \mathrm{St}_n(R) \) is the group generated by the symbols

\[\{ E_{ij}(r) : 1 \leq i \neq j \leq n,\, r \in R \}\]

subject to the following relations:

1.  \( E_{ij}(r) E_{ij}(s) = E_{ij}(r + s) \)
2.  \( [E_{ij}(r), E_{kl}(s)] = 1 \quad \text{if } i \neq l,\, k \neq j \)
3.  \( [E_{ij}(r), E_{jk}(s)] = E_{ik}(rs) \quad \text{if } i \neq k \)

Let $\mathrm{EL}_n(R)$ denote the group generated by elementary matrices over $R$. There is a natural surjective homomorphism

\[\pi_{\mathrm{st}} : \mathrm{St}_n(R) \to \mathrm{EL}_n(R)\]

given by \( \pi_{\mathrm{st}}(E_{ij}(r)) = e_{ij}(r) \). As in the case of \( \mathrm{EL}_n(R) \), if \( \{ x_0 = 1, x_1, \ldots, x_d \} \) is a generating set for \( R \), then \( \mathrm{St}_n(R) \) is generated by the set

\[\mathrm{st} = \{ E_{ij}(x_m) : i, j \in \{1, \ldots, n\},\, i \neq j,\, 0 \leq m \leq d \}.\]

Once we show Property (T) for the Steinberg group, we will get it for $\mathrm{EL}_n(R)$, and in many rings of interest $\mathrm{EL}_n(R)=\mathrm{GL}_n(R)$, including $R=\mathbb{Z}$. In what follows, we will assume $R$ is a finitely generated associative ring.

Definition: Let \( G \) be a group generated by a collection of 6 subgroups \( \{X_{ij} \mid 1 \leq i,j \leq 3,\ i \neq j\} \) such that for any permutation \( i,j,k \) of the set \( \{1,2,3\} \), the following conditions hold:

•  \( X_{ij} \) is abelian;
•  \( X_{ij} \) and \( X_{ik} \) commute;
• \( X_{ji} \) and \( X_{ki} \) commute;
•  \( [X_{ij}, X_{jk}] = X_{ik} \).

Then we will say that \( (G, \{X_{ij}\}) \) is an \( A_2 \)-system. The group \( G \) itself will be called an \( A_2 \)-group.

Let \( Y \) be the graph with 6 vertices \( \{(i,j) : 1 \leq i \neq j \leq 3\} \), such that \( (i,j) \) is connected to \( (k,\ell) \) if and only if \( \{i,j,k,\ell\} = \{1,2,3\} \). Each \( A_2 \)-system \( (G, \{X_{ij}\}) \) has a natural decomposition over \( Y \):

If \( \{i,j,k\} = \{1,2,3\} \), we define the vertex group

\[G(i,j) = \langle X_{ik}, X_{kj} \rangle.\]

Henceforth we will write \( G_{ij} \) for \( G(i,j) \). Note that \( G_{ij} \) is a nilpotent group of class two and \( [G_{ij}, G_{ij}] = X_{ij} \).

The edge groups are defined as follows. If \( e \in E(Y) \) connects \( (i,j) \) and \( (i,k) \), we set \( G_e = \langle X_{ij}, X_{ik} \rangle \). If \( e \in E(Y) \) connects \( (j,i) \) and \( (k,i) \), we set \( G_e = \langle X_{ji}, X_{ki} \rangle \). Finally, if \( e \in E(Y) \) connects \( (i,j) \) and \( (j,k) \), we set \( G_e = X_{ik} \).

Theorem 4 Let \( (G, \{X_{ij}\}) \) be an \( A_2 \)-system, and let \( G_{ij} \) be defined as above. Then

\[\kappa(G, G_{ij}) \geq \frac{3}{8} \quad \text{and} \quad \kappa(G, X_{ij}) \geq \frac{1}{8}.\]

The proof of this is rather technical, but involves careful manipulation of various operators on Hilbert spaces in the style of previous results. We will quote one more proposition:

Proposition 5: For \( 1 \leq i,j \leq 3 \), let \( X_{ij} = \langle e_{kl}(r) : k \in I_i,\ l \in I_j,\ r \in R \rangle \) be the subgroup of \( \mathrm{St}_n(R) \) corresponding to \( X_{ij} \). Then

\[\kappa_r\left( \mathrm{St}_n(R), \bigcup_{i,j} X_{ij};\ \mathcal{S}t \right) \geq \frac{1}{12 \sqrt{2d} + 2\sqrt{3n} + 36\sqrt{2}}.\]

Proof that the Steinberg group has Property (T): We fix a generating set \( \{x_0, x_1, \ldots, x_d\} \) for \( R \), where \( x_0 = 1 \). By definition the set

\[\mathcal{S} := \{ E_{ij}(x_m) : i,j \in \{1, \ldots, n\},\ i \neq j,\ 0 \leq m \leq d \}\]

generates \( \mathrm{St}_n(R) \). For each \( i \neq j \in \{1,2,3\} \), we define the subgroup \( X_{ij} \) of \( \mathrm{St}_n(R) \) by

\[X_{ij} = \langle E_{kl}(r) : k \in I_i,\ l \in I_j,\ r \in R \rangle.\]One checks that \( (\mathrm{St}_n(R), \{X_{ij}\}) \) is an \( A_2 \)-system, and so by Theorem 4 we have

\[\kappa\left( \mathrm{St}_n(R), \bigcup_{i,j} X_{ij} \right) \geq \frac{1}{8}. \]

We conclude using proposition 5. $\blacksquare$

There is another spectral criterion due to Zuk (to be discussed in a later post), which Ershov and Jaikin-Zapirain remark is similar in flavour but appears to apply to different groups. They ask whether there are any connections beyond the superficial similarity. Ershov, Jaikin-Zapirain, and Kassabov later manage to push these techniques even further to show that the Steinberg group over a finitely generated commutative rings $R$ has property (T) for groups graded over any root system. Another application of these techniques is to construct lots of Golod-Shafarevich groups with Property (T) (again to be discussed in a later post).