Averaged projections, angles between groups and strengthening of Banach property (T)

Abstract

Recently, Lafforgue introduced a new strengthening of Banach property (T), which he called strong Banach property (T) and showed that this property has implications regarding fixed point properties and Banach expanders. In this paper, we introduce a new strengthening of Banach property (T), called “robust Banach property (T)”, which is weaker than strong Banach property (T), but is still strong enough to ensure similar applications. Using the method of averaged projections in Banach spaces and introducing a new notion of angles between projections, we establish a criterion for robust Banach property (T) and show several examples of groups in which this criterion is fulfilled. We also derive several applications regarding fixed point properties and Banach expanders and give examples of these applications.

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Acknowledgments

The author would like to thank Mikael de la Salle, Mikhail Ershov and Simeon Reich for reading a previous draft of this paper and adding their insightful comments. Most of the work was done while the author was a visiting assistant professor at the Ohio State University and the author thanks the University for its hospitality.

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Correspondence to Izhar Oppenheim.

Appendix: Applications of robust Banach property (T)

Appendix: Applications of robust Banach property (T)

In this appendix, we’ll prove the applications of robust Banach property (T) for fixed point properties and for Banach expanders. In both cases the proofs are just minor adaptations of the proofs of Lafforgue in [11].

Fixed point property application

We shall prove the following:

Proposition 5.9

Let X be a Banach space and let G be a locally compact group. If G has robust property (T) with respect to \(\mathbb {C} \oplus X\) with the \(l_2\) norm, then any affine isometric action of G on X has a fixed point.

Proof

Let \(\rho \) be an isometric action of G on X. Let \(\overline{0} \in X\) be the zero of (the underlying vector space of) X and define a length l over G as

$$\begin{aligned} l(g) = \max \lbrace \Vert \rho (g).\overline{0} \Vert , 1 \rbrace . \end{aligned}$$

G has robust property (T) with respect to \(\mathbb {C} \oplus X\) and therefore there is some \(s_0 >0\) and a sequence of positive symmetric real functions \(f_n \in C_c (G)\) with \(\int f_n =1\) such for every representation \(\pi \) of G on \(\mathbb {C} \oplus X\) and for any \(0 \le s \le s_0\), if \(\Vert \pi (g) \Vert \le e^{s l (g)}\), then \(\pi (f_n)\) converges to \(\pi (p)\) that is a projection on \((\mathbb {C} \oplus X )^\pi \).

Fix \(D>1\) to be a constant whose value will be determined later and define a representation \(\pi \) as on \(\mathbb {C} \oplus X\) as follows: \(\pi \) is the unique representation on \( \mathbb {C} \oplus X\) such that

$$\begin{aligned} \forall g \in G, \forall v \in X, \pi (g). (D,v) = (D,\rho (g).v). \end{aligned}$$

In other words, \(\pi \) is the representation that keeps \( D \oplus X\) invariant and acts on it via \(\rho \). The reader should note that the subspace \(0 \oplus X\) is pointwise fixed by the action of \(\pi \). Next, we’ll show that

$$\begin{aligned} \Vert \pi (g) \Vert \le 1 + \sqrt{\dfrac{3}{D}} l(g), \forall g \in G. \end{aligned}$$

Indeed,

$$\begin{aligned} \dfrac{\Vert \pi (g).(D,v) \Vert ^2}{\Vert (D,v) \Vert ^2}&= \dfrac{D^2 + \Vert \rho (g). v \Vert ^2}{D^2 + \Vert v \Vert ^2} \le \dfrac{D^2 + (\Vert v \Vert +l(g) )^2}{D^2 + \Vert v \Vert ^2}\nonumber \\&={1 + \dfrac{2 \Vert v \Vert l(g) + l(g)^2}{D^2 + \Vert v \Vert ^2}} \le {1 + \dfrac{2 \Vert v \Vert +1}{D^2 + \Vert v \Vert ^2} l(g)^2}. \end{aligned}$$
(6)

Note that if \(\Vert v \Vert \ge D\), then

$$\begin{aligned} \dfrac{2 \Vert v \Vert +1}{D^2 + \Vert v \Vert ^2} \le \dfrac{3 \Vert v \Vert }{\Vert v \Vert ^2} \le \dfrac{3}{D}. \end{aligned}$$

On the other hand, if \(\Vert v \Vert < D\), then

$$\begin{aligned} \dfrac{2 \Vert v \Vert +1}{D^2 + \Vert v \Vert ^2} \le \dfrac{3 D}{D^2} = \dfrac{3}{D}. \end{aligned}$$

Therefore, we have that for all v that

$$\begin{aligned} \dfrac{2 \Vert v \Vert +1}{D^2 + \Vert v \Vert ^2} \le \dfrac{3}{D}. \end{aligned}$$

Combined with (6), this yields

$$\begin{aligned} \dfrac{\Vert \pi (g).(D,v) \Vert ^2}{\Vert (D,v) \Vert ^2} \le 1 + \dfrac{3}{D} l(g)^2, \end{aligned}$$

and therefore

$$\begin{aligned} \dfrac{\Vert \pi (g).(D,v) \Vert }{\Vert (D,v) \Vert } \le 1 + \sqrt{ \dfrac{3}{D} }l(g). \end{aligned}$$

The above inequality implies that

$$\begin{aligned} \Vert \pi (g) \Vert \le 1 + \sqrt{\dfrac{3}{D}} l(g),\quad \forall g \in G, \end{aligned}$$

as needed. By choosing D large enough, we can therefore insure that we’ll have

$$\begin{aligned} \Vert \pi (g) \Vert \le 1 + s_0 l(g) \le e^{s_0 l(g)}, \quad \forall g \in G. \end{aligned}$$

Therefore \(\pi \) meets the condition for robust Banach property (T) for \(\mathbb {C} \oplus X\). Let \(\lbrace f_n \rbrace \) be the sequence as in the definition of robust Banach property (T). Note that for every n and every \(v \in X\), \(\pi (f_n). (D,v) \in D \oplus X\), since for every n, \(\int f_n =1\). Fix some \(v \in X\) and note that \(\pi (p). (D,v)= \lim _n \pi (f_n).v \in D \oplus X\) and therefore there is some \(v_0 \in X\) such that \(\pi (p). (D,v) = (D,v_0)\). By the definition of \(\pi \), \(v_0\) is a fixed point of the action of G on X through \(\rho \) and we are done. \(\square \)

Banach expanders application

We shall prove the following:

Proposition 5.10

Let G be a finitely generated infinite discrete group with a generating set S and let \(\lbrace N_i \rbrace _{i \in \mathbb {N}}\) be a sequence of finite index normal subgroups of G such that \(S^2 \cap N_i = \lbrace 1 \rbrace \) for every i and \(\vert [G:N_i] \vert \rightarrow \infty \). Let \(\mathcal {E}\) be a class of Banach spaces that is closed under \(l_2\) sums. Fix S to be some symmetric generating set of G. If G has robust Banach property (T) with respect to \(\mathcal {E}\), then the family of Cayley graphs \(\lbrace (G/N_i,S) \rbrace _{i \in \mathbb {N}}\) is a family of X-expanders for any \(X \in \mathcal {E}\).

We shall start by proving the following:

Proposition 5.11

Let G be a discrete finitely generated group with a symmetric generating set S and let \(N_i\) be a sequence of finite index normal subgroups of G such that \(S^2 \cap N_i = \lbrace 1 \rbrace \) for every i and \(\vert [G:N_i] \vert \rightarrow \infty \). Denote by \((V_i,E_i)\) the Cayley graph of \(G / N_i\) with respect to S. Let \(\mathcal {E}\) be a class of Banach spaces such that \(\mathcal {E}\) is closed under \(l_2\) sums. Assume that G has robust Banach property (T) with respect to \(\mathcal {E}\), then there is a constant C such that for every \(X \in \mathcal {E}\), every i and every map \(\phi :(V_i, E_i) \rightarrow X\), we have that there is some \(v (\phi ) \in X\) such that

$$\begin{aligned} \sum _{x \in V_i} \Vert \phi (x) - v (\phi ) \Vert ^2_X \le C \sum _{\lbrace x, y \rbrace \in E_{i}} \Vert \phi (x) - \phi (y) \Vert ^2_X. \end{aligned}$$

Proof

Fix some \(X \in \mathcal {E}\) and some i. Consider \(L^2 (G / N_i ; X)\) with the representation \(\pi : G \rightarrow L^2 (G / N_i ; X)\), defined as \(\pi (g). \phi (g') = \phi (g' g)\). Then \(L^2 (G / N_i ; X)\) is the \(l_2\) sum of \([G:N_i]\) copies of X and therefore \(L^2 (G / N_i ; X) \in \mathcal {E}\). Note that \(\pi \) is an isometric representation on \(L^2 (G / N_i ; X)\) and therefore \(\pi \in \mathcal {F} (\mathcal {E}, 0)\).

From the fact that G has robust Banach property (T) on \(\mathcal {E}\), we get that there is \(p \in C_{\mathcal {F} (\mathcal {E},0)}\) such that \(\pi (p)\) is the projection on \( L^2 (G / N_i ; X)^\pi \). Note that the space of invariant vectors under \(\pi \) is exactly the space of constant functions, so for every \(\phi \in L^2 (G / N_i ; X)\), we can define \(v (\phi ) \in X\) as the constant value of \(\pi (p).\phi \). By the definition of robust Banach property (T) there is a real function \(f \in C_c (G)\) such that \(\int f =1\) and \(\Vert p - f \Vert _{\mathcal {F} (\mathcal {E},0)} \le \frac{1}{2}\).

Note that for every \(\phi \) we have that \(\pi (f) \pi (p). \phi = \pi (p) . \phi \). Using this equality we get that \((\pi (f)-\pi (p)). \phi = (\pi (f)-\pi (p)).(\phi - \pi (p).\phi )\). Therefore

$$\begin{aligned}&\Vert \phi - \pi (p).\phi \Vert _{L^2 (G/N_i ; X)}\\&\quad \le \Vert \phi - \pi (f).\phi \Vert _{L^2 (G/N_i ; X)} + \Vert \pi (f).\phi - \pi (p).\phi \Vert _{L^2 (G/N_i ; X)}\\&\quad = \Vert \phi - \pi (f).\phi \Vert _{L^2 (G/N_i ; X)} + \Vert (\pi (f)-\pi (p)).(\phi - \pi (p).\phi ) \Vert _{L^2 (G/N_i ; X)}\\&\quad \le \Vert \phi - \pi (f).\phi \Vert _{L^2 (G/N_i ; X)} + \frac{1}{2} \Vert \phi - \pi (p).\phi \Vert _{L^2 (G/N_i ; X)}. \end{aligned}$$

This yields that

$$\begin{aligned} \Vert \phi - \pi (p).\phi \Vert _{L^2 (G/N_i ; X)} \le 2 \Vert \phi - \pi (f).\phi \Vert _{L^2 (G/N_i ; X)} \end{aligned}$$

or equivalently

$$\begin{aligned} \Vert \phi - \pi (p).\phi \Vert _{L^2 (G/N_i ; X)}^2 \le 4 \Vert \phi - \pi (f).\phi \Vert _{L^2 (G/N_i ; X)}^2. \end{aligned}$$

Note that by definition

$$\begin{aligned} \Vert \phi - \pi (p).\phi \Vert _{L^2 (G/N_i ; X)}^2 = \sum _{g \in G/N_i} \Vert \phi (g) - v (\phi ) \Vert ^2_X = \sum _{x \in V_i} \Vert \phi (x) - v (\phi ) \Vert ^2_X. \end{aligned}$$

Therefore

$$\begin{aligned} \sum _{x \in V_i} \Vert \phi (x) - v (\phi ) \Vert ^2_X \le 4 \Vert \phi - \pi (f).\phi \Vert _{L^2 (G/N_i ; X)}^2. \end{aligned}$$
(7)

Recall that f is compactly supported and therefore there is some \(k \in \mathbb {N}\) such that \(supp (f) \subseteq S^k\). Note that for every \(g \in S^k\), we have by the triangle inequality that

$$\begin{aligned} \Vert \phi - \pi (g).\phi \Vert _{L^2 (G/N_i ; X)} \le k \sum _{s \in S} \Vert \phi - \pi (s).\phi \Vert _{L^2 (G/N_i ; X)} . \end{aligned}$$

Therefore if we denote \(M = \max _{g \in G} \vert f(g) \vert \), we get that

$$\begin{aligned} \Vert \phi - \pi (f).\phi \Vert _{L^2 (G/N_i ; X)} \le M k \sum _{s \in S} \Vert \phi - \pi (s).\phi \Vert _{L^2 (G/N_i ; X)} . \end{aligned}$$

This yields that

$$\begin{aligned} \Vert \phi - \pi (f).\phi \Vert _{L^2 (G/N_i ; X)}^2&\le \left( M k \sum _{s \in S} \Vert \phi - \pi (s).\phi \Vert _{L^2 (G/N_i ; X)} \right) ^2\\&\le M^2 k^2 \vert S \vert \sum _{s \in S} \Vert \phi - \pi (s).\phi \Vert _{L^2 (G/N_i ; X)}^2\\&= M^2 k^2 \vert S \vert \sum _{g \in G / N_i} \sum _{s \in S} \Vert \phi (g) - \pi (s).\phi (g) \Vert _X^2\\&= M^2 k^2 \vert S \vert \sum _{g \in G / N_i} \sum _{s \in S} \Vert \phi (g) - \phi (gs) \Vert _X^2\\&= M^2 k^2 \vert S \vert \sum _{x \in V_i} \sum _{y \in V_i, \lbrace x,y \rbrace \in E_i} \Vert \phi (x) - \phi (y) \Vert _X^2\\&= 2 M^2 k^2 \vert S \vert \sum _{\lbrace x, y \rbrace \in E_i } \Vert \phi (x) - \phi (y) \Vert _X^2. \end{aligned}$$

We are done by combining the above with (7). Note that \(M,k, \vert S \vert \) are all independent of the choice of \(N_i\) and therefore taking \(C = 8 M^2 k^2 \vert S \vert \) gives a constant that is uniform for all \(N_i\)’s. \(\square \)

Using this proposition, we can prove Proposition 5.10, by proving the following lemma:

Lemma 5.12

Let X be a Banach space and let \(\lbrace (V_i,E_i) \rbrace _{i \in \mathbb {N}}\) be a family of graphs with uniformly bounded valency such that \(\vert V_i \vert \rightarrow \infty \) as \(i \rightarrow \infty \). Assume that there is a constant C such that for every \(i \in \mathbb {N}\) and every \(\phi : (V_i,E_i) \rightarrow X\) there is some \(v (\phi ) \in X\) such that

$$\begin{aligned} \sum _{x \in V_i} \Vert \phi (x) - v (\phi ) \Vert ^2_X \le C \sum _{\lbrace x, y \rbrace \in E_{i}} \Vert \phi (x) - \phi (y) \Vert ^2_X. \end{aligned}$$

Then \(\lbrace (V_i,E_i) \rbrace _{i \in \mathbb {N}}\) is a family of X-expanders.

Proof

Let \(D \ge 2\) be a uniform bound on the valency of \(\lbrace (V_i,E_i) \rbrace _{i \in \mathbb {N}}\). Assume towards contradiction that there is a sequence of maps \(\phi _i : V_i \rightarrow X\) and functions \(\rho _-, \rho _+ : \mathbb {N} \rightarrow \mathbb {R}\) such that \(\lim _k \rho _- (k) = \infty \) and

$$\begin{aligned} \forall i \in \mathbb {N}, \forall x,y \in V_i, \rho _{-} (d_i (x,y)) \le \Vert \phi _i (x) - \phi _i (y) \Vert \le \rho _+ (d_i (x,y)), \end{aligned}$$

where \(d_i (x,y)\) is the distance in the graph \((V_i,E_i)\) between x and y. By replacing \(\phi _i\) by \(\phi _i - v (\phi _i)\), we can assume that for every such \(\phi _i\), we have that

$$\begin{aligned} \sum _{x \in V_i} \Vert \phi _i (x) \Vert ^2_X \le C \sum _{\lbrace x, y \rbrace \in E_{i}} \Vert \phi _i (x) - \phi _i (y) \Vert ^2_X. \end{aligned}$$

Note that

$$\begin{aligned} \sum _{\lbrace x, y \rbrace \in E_{i}} \Vert \phi _i (x) - \phi _i (y) \Vert ^2_X \le \vert E_i \vert \rho _+ (1)^2 \le \dfrac{D \vert V_i \vert }{2} \rho _+ (1)^2. \end{aligned}$$

Therefore

$$\begin{aligned} \sum _{x \in V_i} \Vert \phi _i (x) \Vert ^2_X \le \frac{\vert V_i \vert }{2} (C D \rho _+ (1)^2). \end{aligned}$$

Consider the median value of the multiset \(\lbrace \Vert \phi _i (x) \Vert : x \in V_i \rbrace \). If this median is strictly greater than \(\sqrt{C D} \rho _+ (1)\), we get a contradiction to the above inequality. Therefore, there is a set \(U_i \subseteq V_i\) such that \(\vert U_i \vert \ge \lfloor \frac{\vert V_i \vert }{2} \rfloor \) and

$$\begin{aligned} \forall x \in U_i, \Vert \phi _i (x) \Vert \le \sqrt{C D} \rho _+ (1). \end{aligned}$$

Therefore by triangle inequality

$$\begin{aligned} \forall x,y \in U_i, \Vert \phi _i (x) - \phi _i (y) \Vert \le 2\sqrt{CD} \rho _+ (1). \end{aligned}$$
(8)

On the other hand, since the valency in all the graphs is bounded by D, we have that

$$\begin{aligned} \forall i \in \mathbb {N}, \forall k \in \mathbb {N}, \forall x \in V_i, \vert \lbrace y \in V_i : d_i (x,y) < k \rbrace \vert < D^{k}. \end{aligned}$$

Denote \(diam (U_i)\) to be the diameter of \(U_i\) in \(V_i\), then by the above inequality we get that

$$\begin{aligned} diam (U_i) \ge \frac{\ln (\vert U_i \vert )}{\ln (D)} \ge \frac{\ln (\lfloor \frac{\vert V_i \vert }{2} \rfloor )}{\ln (D)}. \end{aligned}$$

Therefore there are \(x,y \in U_i\) such that

$$\begin{aligned} \rho _{-} \left( \frac{\ln (\lfloor \frac{\vert V_i \vert }{2} \rfloor )}{\ln (D)} \right) \le \Vert \phi _i (x) - \phi _i (y) \Vert . \end{aligned}$$

Combining this with (8) yields that for every i we have that

$$\begin{aligned} \rho _{-} \left( \frac{\ln (\lfloor \frac{\vert V_i \vert }{2} \rfloor )}{\ln (D)} \right) \le 2\sqrt{CD} \rho _+ (1). \end{aligned}$$

But from the assumption that \(\lim _i \vert V_i \vert = \infty \) we get a contradiction to the assumption that \(\lim _k \rho _- (k) = \infty \). \(\square \)

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Oppenheim, I. Averaged projections, angles between groups and strengthening of Banach property (T). Math. Ann. 367, 623–666 (2017). https://doi.org/10.1007/s00208-016-1413-2

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Mathematics Subject Classification

  • 20F99
  • 46B85