3.1 Simple convergence

Before considering issues related to norm convergence, let us note that it implies simple convergence. We should therefore first understand under which condition the sequence $\varphi ^{k}$ converges simply to $\delta _{e}$ . For compact groups, it is known that a random walk converges to the Haar state if and only if its support is not contained in a closed subgroup or in a coset with respect to a normal subgroup (see for instance [Reference Stromberg19, Thm 3.2.4]). For general compact (and even finite) quantum groups, generalizing this equivalence is still an open problem. In our case however, we can settle it. Let us first give a definition for convenience:

It is clear that simple convergence to $\delta _{e}$ is equivalent to the initial function being strict. A typical example of a nonstrict positive definite function is the counit $\varepsilon : \Gamma \to \mathbb {C}$ sending each g to $1$ . More generally, any character (i.e. one-dimensional representation) of $\Gamma $ is not strict. The next result says that this is basically the only obstruction.

Proof $(1)\Rightarrow (2)$ : Assume that $\varphi $ is not strict and set

$$\begin{align*}\Lambda = \{h\in \Gamma\mid \vert\varphi(h)\vert = 1\}. \end{align*}$$

The GNS construction provides a unitary representation $\pi : \Gamma \to B(H)$ on a Hilbert space H and a unit vector $\xi \in H$ such that for all $g\in \Gamma $ , $\varphi (g) = \langle \pi (g)\xi , \xi \rangle $ . For $h\in \Lambda $ , the Cauchy–Schwarz inequality being an equality, $\pi (h)\xi $ is colinear to $\xi $ , hence $\pi (h)\xi = \varphi (h)\xi $ . Thus, for any $g\in \Gamma $ ,

$$\begin{align*}\vert\varphi(gh)\vert = \vert\langle\pi(gh)\xi, \xi\rangle\vert = \vert \varphi(h)\langle\pi(g)\xi, \xi\rangle\vert = \vert\varphi(g)\vert. \end{align*}$$

As a first consequence, $\Lambda $ is stable under multiplication. Since moreover $\varphi (g^{-1}) = \overline {\varphi (g)}$ for any $g\in \Gamma $ , we conclude that $\Lambda $ is a subgroup. Moreover, $\varphi : \Lambda \to \mathbb {C}$ is a character and for any $h\in \Lambda $ and $g\in \Gamma $ , $\varphi (gh) = \varphi (g)\varphi (h) = \varphi (hg)$ .

$(2)\Rightarrow (3)$ : Considering again the GNS representation of $\varphi $ , the assertion is equivalent to the fact that $\mathbb {C}\xi $ is globally invariant under the action of $\Lambda $ , from which the bimodularity follows.

$(3)\Rightarrow (1)$ : If $\varphi $ is $\Lambda $ -bimodular, then for any $h\in \Lambda $ we have

$$\begin{align*}\vert\varphi(h)\vert^{2} = \overline{\varphi(h)}\varphi(h) = \varphi(h^{-1})\varphi(h) = \varphi(e) = 1 \end{align*}$$

so that $\varphi $ is not strict. ▪

Remark 3.2 If $\Gamma $ is abelian, then there exists a probability measure $\mu _{\varphi }$ on the dual compact abelian group $\hat {\Gamma }$ such that for all $g\in \Gamma $ ,

$$\begin{align*}\varphi(g) = \int_{\hat{\Gamma}}\chi(g)\mathrm{d}\mu_{\varphi}(\chi). \end{align*}$$

The second condition in Proposition 3.1 yields a character $\eta \in \hat {\Lambda }$ such that $\varphi _{\vert \Lambda } = \eta $ ; hence for any $h\in \Lambda $ ,

$$\begin{align*}\int_{\hat{\Gamma}}(\eta^{-1}\chi)(h)\mathrm{d}\mu_{\varphi}(\chi) = 1. \end{align*}$$

This implies that the support of $\mu $ is contained in $\eta \Lambda ^{\perp }$ , where

$$\begin{align*}\Lambda^{\perp} = \{\chi\in \hat{\Gamma} \mid \Lambda\subset\ker(\chi)\} \end{align*}$$

is the annihilator of $\Lambda $ . Since any subgroup is normal in the abelian case, we recover the classical criterion.

3.2 Absolute continuity

As we have seen in the proof, Lemma 2.1 is useful only when $\varphi $ has an $L^{1}$ -density with respect to $\delta _{e}$ , which by [Reference Takesaki20, Thm V.2.18] is equivalent to the fact that $\varphi $ extends to a normal linear map on $L(\Gamma )$ . Moreover, as the proof of Lemma 2.1 shows, it is more practical to consider $L^{2}$ -densities, since this reduces the problem to the convergence of series in $\ell ^{2}(\Gamma )$ . The latter problem naturally involves the rate of decay of $\varphi $ and to make this more precise we introduce the following quantities:

$$\begin{align*}\varphi^{+}(i) = -\inf_{g\in S(i)}\ln(\vert \varphi(g)\vert) \text{ and } \varphi^{-}(i) = -\mathop{\mathrm{sup}}\limits_{g\in S(i)}\ln(\vert \varphi(g)\vert). \end{align*}$$

The definition may seem unnatural but is designed to fit with growth conditions for cocycles, which will be our main source of examples in Section 4.

Proof We start from the straightforward inequalities

$$\begin{align*}\sum_{i=0}^{+\infty}s_{i} e^{-2k\varphi^{+}(i)}\leqslant \sum_{g\in \Gamma}\vert\varphi(g)\vert^{2k} \leqslant \sum_{i=0}^{+\infty}s_{i} e^{-2k\varphi^{-}(i)}. \end{align*}$$

For the right-hand side, the Cauchy radical test gives a sufficient condition for convergence, namely

$$\begin{align*}\limsup_{i}s_{i}^{1/i}e^{-2k\varphi^{-}(i)/i} < 1. \end{align*}$$

Because $s_{i}^{1/i}$ converges to $\omega (S)$ , this gives the first part of the statement. On the other hand, if the series in the middle converges, then so does the one on the left-hand side and using the Cauchy radical test again yields the second part of the statement. ▪

Proposition 3.3 settles the problem of the existence of $L^{2}$ -densities, which is what we need in order to apply Lemma 2.1. However, it can be that a state $\varphi $ has an $L^{1}$ -density without having an $L^{2}$ -density. We will now give a criterion for absolute continuity for some particular class of groups. The idea will be to show that $\varphi ^{k}$ is not bounded on $L(\Gamma )$ if k is too small by evaluating it on suitable test functions. If a length function on $\Gamma $ is fixed, the natural candidates are the elements

$$\begin{align*}\chi_{i} = \sum_{g\in S(i)} g. \end{align*}$$

Doing this requires a control on the norm of these elements in $L(\Gamma )$ in terms of the sizes of the spheres and such a control is given by the Property of Rapid Decay [Reference Jolissaint16]. We, however, need an extra positivity assumption on $\varphi $ .

Proposition 3.4 Let $\Gamma $ be a discrete group of exponential growth with the Property of Rapid Decay and let $\varphi $ be a positive definite function on $\Gamma $ taking only positive values. Then, $\varphi ^{k}$ extends to a bounded normal functional on $L(\Gamma )$ only if

$$\begin{align*}\liminf_{i}\frac{\varphi^{+}(i)}{i} \geqslant \frac{\ln(\omega(S))}{2k}. \end{align*}$$

Proof The Property of Rapid Decay provides a polynomial P such that for any element $x\in S(i)$ , $\|x\|_{\infty }\leqslant P(i)\|x\|_{2}$ . Thus, since $\|\chi _{i}\|_{2} = \sqrt {s_{i}}$ ,

$$\begin{align*}\frac{\vert\varphi^{k}(\chi_{i})\vert}{\|\chi_{i}\|_{\infty}}\geqslant \frac{1}{P(i)}e^{\frac{\ln(s_{i})}{2}-k\varphi^{+}(i)}. \end{align*}$$

If the left-hand side is bounded, then there exists $C> 0$ such that for i large enough $\frac {\ln (s_{i})}{2}-k\varphi _{i}^{+}\leqslant C$ . Then,

$$\begin{align*}k\frac{\varphi^{+}(i)}{i}\geqslant \frac{\ln(s_{i})}{2i} - \frac{C}{i}. \end{align*}$$

and because $\Gamma $ is assumed to have exponential growth, the result follows. ▪

Recall that an amenable group has the Property of Rapid Decay if and only if it has polynomial growth by [Reference Jolissaint16, Cor 3.1.8]. The previous proposition is therefore useful only for nonamenable groups.

3.3 Estimates

In view of the results of the previous section, in order to be able to use Lemma 2.1 we need to make an assumption on the rate of decay of $\varphi $ . Proposition 3.3 suggests the condition $\liminf _{i}\varphi ^{+}(i)/i> 0$ but we will need a stronger one:

Definition 3.2 Let $\Gamma $ be a discrete group and let $\varphi $ be a positive definite function on $\Gamma $ . It is said to have exponential decay if there exists $\alpha> 0$ and a finite symmetric generating set S such that for all $g\in \Gamma $ ,

$$\begin{align*}\vert\varphi(g)\vert \leqslant e^{-\alpha\vert g\vert_{S}}. \end{align*}$$

Having exponential decay is independent from the choice of a finite symmetric generating set S.

Remark 3.5 One may expect a bound of the form $C_{0}e^{-\alpha \vert g\vert _{S}}$ in the above definition, but if we moreover assume that $\varphi $ is strict then this is equivalent to our definition. Indeed, there exist $n_{0}$ and $\alpha '> 0$ such that for $\vert g\vert _{S}> n_{0}$ , $C_{0}e^{-\alpha \vert g\vert _{S}}\leqslant e^{-\alpha '\vert g\vert _{S}}$ . Moreover, because $\varphi $ is strict,

$$\begin{align*}\alpha'' = \inf_{g\in B(n_{0})\setminus\{e\}}\frac{-\ln(\vert \varphi(g)\vert)}{\vert g\vert_{S}}> 0 \end{align*}$$

and setting $\alpha = \min (\alpha ', \alpha '')$ yields the result. We are therefore including strictness in the definition of exponential decay.

By definition, $\alpha $ is less than $\liminf _{i}\varphi ^{-}(i)/i$ so that the latter is the best potential decay rate, and since it depends on S, we will denote it by $\alpha (S)$ . With this in hand, we can give a general upper bound statement. The condition for the existence of an $L^{2}$ -density suggests that the threshold for exponential convergence should be $\ln (\omega (S))/2\alpha (S)$ . However, because $\omega (S)$ is an infimum we cannot use it to bound the series appearing in Lemma 2.1. We will therefore have to take some room and use $\ln (\vert S\vert - 1)/2\alpha (S)$ instead.

Proposition 3.6 Let $\Gamma $ be a discrete group with finite symmetric generating set S and let $\varphi : \Gamma \to \mathbb {C}$ be a positive definite function with exponential decay. Then, for any $c> 0$ and $k \geqslant \ln (\vert S\vert -1)/2\alpha (S) + c$ ,

$$\begin{align*}\|\varphi^{k} - \delta_{e}\| \leqslant \frac{e^{-\alpha(S) c}}{\sqrt{2 - 2e^{-\alpha(S) c}}}. \end{align*}$$

Proof Lemma 2.1 yields

$$ \begin{align*} \|\varphi^{k} - \delta_{e}\|^{2} & \leqslant \frac{1}{4}\sum_{g\neq e}\vert\varphi(g)\vert^{2k} \leqslant \frac{1}{4}\sum_{i=1}^{+\infty}s_{i} e^{-2k \alpha(S)i} \\[-3pt] & \leqslant \sum_{i=1}^{+\infty}\frac{\vert S\vert}{4}(\vert S\vert - 1)^{i-1} e^{-2k \alpha(S) i} \\[-3pt] & = \frac{\vert S\vert}{4} e^{-2k\alpha(S)}\frac{1}{1 - (\vert S\vert - 1)e^{-2k\alpha(S)}}. \end{align*} $$

For $k = \ln (\vert S\vert )/2\alpha (S) + c$ we therefore get

$$\begin{align*}\|\varphi^{k} - \delta_{e}\|^{2} \leqslant \frac{1}{4}\frac{\vert S\vert}{(\vert S\vert - 1)}\frac{e^{-2\alpha(S) c}}{1 - e^{-2\alpha(S) c}} \leqslant \frac{e^{-2\alpha(S) c}}{2 - 2e^{-2\alpha(S) c}} \end{align*}$$

and the result follows. ▪

To establish a cut-off phenomenon, it is necessary to have both an upper and a lower bound. Usually, the hard work concerns the upper bound but in our setting we will see that one needs specific arguments involving the particular form of the positive definite functions to prove lower bounds. For the moment we will nevertheless give a general lower bound which only requires $\varphi $ to take positive values (which will always be the case in our examples), but which uses the norm on the von Neumann algebra $L(\Gamma )$ (see below for comments concerning this).

Recall that any finite generating set S of cardinality n gives rise to a quotient map $p : \mathbb {F}_{n}\to \Gamma $ . Let $r_{i}$ be the number of reduced words in $\ker (p)$ of length i. By [Reference Cohen5, Prop 1], $r_{i}^{1/i}$ converges (once vanishing terms are removed) to a number $\gamma (S)$ called the cogrowth rate of S.

Proposition 3.7 Let $\Gamma $ be a discrete group with a finite generating set S and let $\varphi $ be a positive definite function on $\Gamma $ such that $\varphi (g)\geqslant 0$ for all $g\in S$ . Then, for any $c> 0$ and $k \leqslant \ln (\vert S\vert - 1)/2\varphi ^{+}(1) - c$ such that $\varphi ^{k}$ has a bounded extension to $L(\Gamma )$ ,

$$\begin{align*}\|\varphi^{k} - \delta_{e}\| \geqslant 1 - 4\left(3 + \frac{\gamma(S)^{2}}{\vert S\vert}\right)e^{-2\varphi^{+}(1)c}. \end{align*}$$

Proof It is easy to prove (see for instance [Reference Freslon12, Lem 2.6]) that the norm $\|\varphi ^{k} - \delta _{e}\|_{L(\Gamma )^{*}}$ is equal to the supremum of $\vert \varphi ^{k}(p) - \delta _{e}(p)\vert $ over all projections $p\in L(\Gamma )$ . Thus, the lower bound will be obtained by evaluating at a suitably chosen spectral projection p of $\chi _{1}$ . Choosing the projection first requires some estimates.

According to [Reference Cohen5, Thm 3], $\|\chi _{1}\|_{\infty } = \gamma (S) + (\vert S\vert - 1)/\gamma (S)$ . This only makes sense for groups which are not free on S, but the formula can be extended to the latter case by setting $\gamma (S) = \sqrt {\vert S\vert - 1}$ instead of $1$ . Since $\chi _{1}$ is self-adjoint, it follows from this that

$$\begin{align*}\operatorname{\mathrm{var}}_{\varphi}(\chi_{1})\leqslant \left(\gamma(S) + \frac{\vert S\vert}{\gamma(S)}\right)^{2}, \end{align*}$$

where $ \operatorname {\mathrm {var}}_{\varphi }(\chi _{1})$ denote the variance of $\chi _{1}$ under $\varphi ;$ that is to say

$$\begin{align*}\operatorname{\mathrm{var}}_{\varphi}(\chi_{1}) = \varphi(\chi_{1}^{2}) - \varphi(\chi_{1})^{2}. \end{align*}$$

On the other hand, with our assumption we have the estimate

$$\begin{align*}\varphi(\chi_{1}) = \sum_{g\in S} \varphi(g) \geqslant \vert S\vert e^{-k\varphi^{+}(1)}.\end{align*}$$

The lower bound can be obtained from this using the same arguments as in [Reference Freslon12, Prop 3.15]. Namely, set $\eta = \vert S\vert e^{-k\varphi ^{+}(1)}$ and let us view $\chi _{1}$ as a classical random variable in the algebra $L^{\infty }(\text {Sp}(\chi _{1}))$ which it generates inside $L(\Gamma )$ . On the one hand, if $\vert \chi _{1}\vert \leqslant \eta /2$ then $\vert \chi _{1} - \varphi (\chi _{1})\vert \geqslant \eta /2$ and the probability, with respect to $\varphi $ , of this event can be bounded by the Chebyshev inequality. On the other hand, the probability, with respect to $\delta _{e}$ , that $\vert \chi _{1}\vert \leqslant \eta /2$ is one minus the probability that $\vert \chi _{1}\vert> \eta /2$ and the latter can also be bounded using the Chebyshev inequality. Putting things together yields

$$ \begin{align*} \|\varphi^{\ast k} - \delta_{e}\| & \geqslant 1 - \frac{4}{\vert S\vert^{2}}\left(\gamma(S) + \frac{\vert S\vert}{\gamma(S)}\right)^{2}e^{2k\varphi^{+}(1)} - \frac{4}{\vert S\vert}e^{2k\varphi^{+}(1)} \\ & \geqslant 1 - 4\left(\frac{\gamma(S)}{\sqrt{\vert S\vert}} + \frac{\sqrt{\vert S\vert}}{\gamma(S)}\right)^{2}e^{-2\varphi^{+}(1)c} - 4e^{-2\varphi^{+}(1)c} \end{align*} $$

and the result follows from the fact that $\gamma (S)\geqslant \sqrt {\vert S\vert }$ (see [Reference Cohen5, Thm 1]). ▪

That statement has two practical issues. The first one is that there is no reason in general that $\varphi ^{k}$ will have a bounded extension to $L(\Gamma )$ for such k. At least, Proposition 3.3 ensures that this will work for

$$\begin{align*}c\leqslant \frac{\ln(\vert S\vert - 1)}{2\liminf \varphi^{-}(i)/i} - \frac{\ln(\omega(S))}{2\varphi^{+}(1)}. \end{align*}$$

However, when considering a sequence of groups, that quantity may go to $0$ .

The second issue is that the lower bound involves a term depending on the size of S. In order to convert this into a cut-off statement, we would need to find families of groups $\Gamma _{N}$ with generating sets $S_{N}$ such that we have a uniform bound on $\gamma (S_{N})^{2}/\vert S_{N}\vert $ . In a sense, this means that they are close to free groups. Note, however, that it is shown in [Reference Ollivier17] that for any $\epsilon> 0$ , groups with a finite generating set S such that $\gamma (S)\leqslant \sqrt {\vert S\vert } + \epsilon $ are generic in the sense of random groups.

Because of these two reasons, we will proceed differently to prove the lower bound in concrete examples, taking advantage of the peculiarities of the positive definite functions under consideration.

4.1 Cut-off

We can now give some examples of positive definite functions yielding a sharp cut-off. For this, we need a family $(\Gamma _{N})_{N\in \mathbb {N}}$ of discrete groups together with $1$ -cocycles $b_{N}$ and symmetric generating sets $S_{N}$ such that

1. $b_{N}$ has radical growth;

2. $\varphi _{b_{N}}^{+}(1) = \alpha (S_{N});$

3. $\gamma (S_{N})/\sqrt {\vert S_{N}\vert }$ is uniformly bounded.

The simplest instance when the first two conditions are met is when $\|b(g)\|^{2} = \vert g\vert _{S}$ , which is possible if and only if (see for instance [Reference Bekka, Harpe and Valette2, Thm C.2.3]) the word length $\vert \cdot \vert _{S}$ is conditionally negative definite [Reference Bekka, Harpe and Valette2, Sec 2.10]. But even in that case, there is no reason why the third condition should be met. Even if it were, Proposition 3.7 may prove useless, since for instance if follows from [Reference Arano, Isono and Marrakchi1, Prop 3.8] that $\varphi _{b_{N}}^{s}$ cannot be bounded on $L(\Gamma )$ for s arbitrarily close to $0$ , so that the values of c for which the lower bound applies are bounded below.

It turns out, however, that we can go round this problem. Note that if quantities are to depend on the size of $S_{N}$ , then we should assume that the generating set is minimal to get optimal constants.

Proof The upper bound comes directly from Proposition 3.6 with $\alpha (S) = 1$ . As for the lower bound, we will prove that for any $c>0$ and $k \leqslant \ln (\vert S_{N}\vert - 1)/2 - c$ ,

$$\begin{align*}\|\varphi_{b_{N}}^{k} - \delta_{e}\| \geqslant 1 - 8e^{-2c}. \end{align*}$$

We will proceed in a similar way as in Proposition 3.7, using $\chi _{1}$ , except that we cannot reason in $L(\Gamma )$ . Let us start by computing the expectation

$$\begin{align*}\varphi_{b_{N}}^{k}(\chi_{1}) = \vert S_{N}\vert e^{-k}. \end{align*}$$

For the variance, note that because a product of two elements of $S_{N}$ has length two unless the elements are inverse to one another (by minimality of $S_{N}$ ), we have

$$\begin{align*}\chi_{1}^{2} = \vert S_{N}\vert e + (\vert S_{N}\vert^{2} - \vert S_{N}\vert)\chi_{2}. \end{align*}$$

Thus,

$$ \begin{align*} \operatorname{\mathrm{var}}_{\varphi_{b_{N}}^{k}}(\chi_{1}) & \leqslant \vert S_{N}\vert + (\vert S_{N}\vert^{2} - \vert S_{N}\vert)e^{-2k} - \vert S_{N}\vert^{2}e^{-2k} \\ & = \vert S_{N}\vert(1-e^{-2k}). \end{align*} $$

Let us now consider the C*-algebra generated by $\chi _{1}$ in $C^{*}(\Gamma _{N})$ . It is commutative because $\chi _{1}$ is self-adjoint, hence by Gelfand duality it is isomorphic to $C(K)$ , where the compact space $K\subset \mathbb {R}$ is the spectrum of $\chi _{1}$ . Let us set $\eta = \vert S_{N}\vert e^{-k}$ and consider the spectral projection of $\chi _{1}$ in $L^{\infty }(K)$ corresponding to the interval $[0, \eta /2]$ . Borel functional calculus asserts that there exists a Borel subset $B\subset K$ such that p is the indicator function of B. Moreover, $\varphi ^{k}$ and $\delta _{e}$ both induce by the Riesz representation theorem Borel probability measures, say $\mu $ and $\nu $ , on K whose total variation distance equals their distance as states on $C(K)$ . The Chebyshev inequality applied to $\mu $ yields

$$ \begin{align*} \mu(B) & = \mathbb{P}\left(\vert \chi_{1}\vert \leqslant \eta/2\right) \\ & \leqslant \mathbb{P}\left(\vert \mathbb{E}_{\mu}(\chi_{1}) - \chi_{1}\vert \geqslant \eta/2\right) \\ & \leqslant \frac{4\operatorname{\mathrm{var}}_{\varphi}(\chi_{1})}{\eta^{2}} \\ & = 4\frac{\vert S_{N}\vert (1-e^{-2k})}{\vert S_{N}\vert^{2}e^{-2k}} \\ & \leqslant 4e^{-2c}. \end{align*} $$

Furthermore, using $\delta _{e}(\chi _{1}) = 0$ and $\delta _{e}(\chi _{1}^{2}) = \vert S_{N}\vert $ , we have

$$ \begin{align*} \nu(B) & = 1 - \nu(K\setminus B) \\ & \geqslant 1 - 4\frac{\vert S_{N}\vert}{\eta^{2}} \\ & \geqslant 1 - 4e^{-2c}. \end{align*} $$

We can now conclude that

$$\begin{align*} \|\varphi^{k} -\delta_{e}\| \geqslant \|\varphi^{k}_{\mid C(K)} - \delta_{e\mid C(K)}\| = \|\mu - \nu\|_{TV} \geqslant \nu(B) - \mu(B) \geqslant 1 - 8e^{-2c}.{\blacksquare} \end{align*}$$

Remark 4.3 The argument above is in fact more general: for any group $\Gamma $ with a minimal symmetric generating set S and a positive definite function $\varphi $ , if $\varphi ^{-}(2)\geqslant 2\varphi ^{+}(1)$ then for any $c> 0$ and $k \leqslant \ln (\vert S\vert - 1)/2\varphi ^{+}(1) - c$ ,

$$\begin{align*}\|\varphi^{k} - h\| \geqslant 1 - 8e^{-2\varphi^{+}(1)c}. \end{align*}$$

Here are two families of examples one can build from this:

1. For $N\in \mathbb {N}$ , let $\Gamma _{N} = \mathbb {F}_{N}$ be the free group on N generators and take for $S_{N}$ the canonical generators and their inverses. By [Reference Haagerup15, Lem 1.2], the associated word length is conditionally negative definite so that for the corresponding state, Theorem 4.2 yields a cut-off at $\ln \left (\sqrt {2N-1}\right )$ steps. Note that in that case, the cut-off parameter is indeed equal to the exponential growth rate $\omega (S_{N})$ . Moreover, free groups have the Property of Rapid Decay by [Reference Haagerup15, Lem 1.4] so that by Proposition 3.4, the cut-off happens exactly when the state extends to a bounded normal functional on the von Neumann algebra.

2. Let $S_{N}$ be a set with N elements and let $W_{N}$ be a Coxeter matrix of size $N\times N$ . If the corresponding Coxeter group $\Gamma _{N}$ is infinite, it follows from [Reference Bozejko, Januszkiewicz and Spatzier3] that the word length associated to $S_{N}$ is conditionally negative definite. Then, by Theorem 4.2 there is a cut-off at $\ln (\sqrt {N-1})$ steps. Moreover, $\varphi _{b_{N}}^{\lfloor \ln (\vert S\vert - 1) - c\rfloor }$ extends to $L(\Gamma _{N})$ at least for $c < \ln ((\vert S\vert - 1)/\omega (S))/2$ . Note also that infinite Coxeter groups satisfy the Property of Rapid Decay by [Reference Fendler11, Cor 1].

4.2 Free products

The examples of $\mathbb {F}_{N} = \mathbb {Z}^{\ast N}$ and $\mathbb {Z}_{2}^{\ast N}$ (this is a Coxeter group) suggest considering more general free products to build sequences of groups exhibiting a cut-off phenomenon. Indeed, there is a natural way to build a free product of two positive definite functions and the growth of the result is easily controlled. Given positive definite functions $\varphi _{1}$ and $\varphi _{2}$ on groups $\Lambda _{1}$ and $\Lambda _{2}$ , any element $g\in \Lambda _{1}\ast \Lambda _{2}$ can be uniquely written as an alternating product $g = h_{1}k_{1}h_{2}k_{2}\cdots h_{n}k_{n}$ with $h_{i}\in \Lambda _{1}$ and $k_{i}\in \Lambda _{2}$ not being the neutral elements. Then, setting

$$\begin{align*}\varphi(g) = \varphi_{1}(h_{1})\varphi_{2}(k_{1})\cdots \varphi_{1}(h_{n})\varphi_{2}(k_{n}) \end{align*}$$

defines a positive definite function (see for instance [Reference Cherix, Cowling, Jolissaint, Julg and Valette4, Prop 6.2.3]). We will denote that function by $\varphi _{1}\ast \varphi _{2}$ . Let us gather all the sufficient conditions to produce a cut-off phenomenon in this context into a single proposition:

Proof Consider an element $g = g_{1}\cdots g_{n}\in \Gamma _{N}$ where $g_{j}\in \Lambda _{i_{j}}$ and $i_{j}\neq i_{j+1}$ . By construction,

$$\begin{align*}\varphi_{N}(g) = \prod_{j=1}^{n}\psi_{i_{j}}(g_{j})\leqslant \prod_{j=1}^{n}e^{-\alpha_{i_{j}}\vert g_{j}\vert_{S_{i_{j}}}} \leqslant \exp\left(-\alpha\displaystyle\sum_{j=1}^{n}\vert g_{j}\vert_{S_{i_{j}}}\right) = e^{-\alpha\vert g\vert} \end{align*}$$

so that we have a uniform control on the exponential growth rate. Moreover, because elements of length one are exactly generators of the initial groups,

$$\begin{align*}\varphi_{N}^{+}(1) = \mathop{\mathrm{sup}}\limits_{1\leqslant i\leqslant N}\psi_{i}^{+}(1)\leqslant \beta. \end{align*}$$

To be able to conclude we now need a lower bound which can be obtained by a free probability argument. More precisely, we can consider the elements $\chi _{1}^{(i)}$ as noncommutative random variables in the noncommutative probability space $(C^{*}(\Gamma _{N}), \varphi _{N})$ . Because $\varphi _{N}$ is a free product state, the aforementioned variables are freely independent with respect to it. Thus, the variance of their sum is the sum of their variances and

$$\begin{align*}\operatorname{\mathrm{var}}_{\varphi_{N}}(\chi_{1}) = \sum_{i=1}^{N}\operatorname{\mathrm{var}}_{\psi_{i}}(\chi_{1}^{(i)}) \leqslant \delta\sum_{i=1}^{N}\vert T_{i}\vert = \delta\vert S_{N}\vert. \end{align*}$$

It now follows, as in the proof of Theorem 4.2, that for $k\leqslant \ln (\vert S_{N}\vert - 1)/2\beta - c$ ,

$$\begin{align*} \|\varphi_{N}^{k} - \delta_{e}\| \geqslant 1 - 4(\delta + 1)e^{-2\beta c}. {\blacksquare}\end{align*}$$

The simplest instance where the hypothesis of this proposition is satisfied is when the sequence is constant, i.e. we consider $\varphi ^{\ast N}$ on $\Gamma ^{\ast N}$ with generating set $S^{\sqcup N}$ . All we need is then a positive definite function with exponential decay, or a cocycle with radical growth. Examples of such cocycles will be given in the next subsection.

4.3 Geometric cocycles

We will now give examples of $1$ -cocycles with radical growth on discrete groups obtained by geometric means. To this purpose, let us say that a metric space $(X, d)$ has an equivariant Hilbert embedding with radical growth if there exists

1. A Hilbert space H together with an affine isometric action of the isometry group of X;

2. An equivariant map $f : X\to H$ such that

   $$\begin{align*}C^{-1}\sqrt{d(x, y)} - C'\leqslant \|f(x) - f(y)\|\leqslant Cd(x, y) + C' \end{align*}$$
   for some constants $C> 0$ and $C'\geqslant 0$ .

Recall that a group $\Gamma $ is said to act geometrically on a metric space $(X, d)$ if it acts properly and cocompactly by isometries. Here is a well-known recipe to produce $1$ -cocycles with radical growth:

Proof By the Švarc–Milnor Lemma (see for instance [Reference Harpe6, Thm 23]), $\Gamma $ is quasi-isometric to $(X, d)$ so that composing with the equivariant embedding of X we get a $1$ -cocycle b on $\Gamma $ satisfying

(4.1) $$ \begin{align} \|b(g)\|^{2} \geqslant C_{0}\vert g\vert_{S} - C_{1} \end{align} $$

for all $g\in \Gamma $ , with constants $C_{0}> 0$ and $C_{1}\geqslant 0$ . Set $\varphi _{b}(g) = e^{-\|b(g)\|^{2}}$ and

$$\begin{align*}\Lambda = \{g\in \Gamma \mid b(g) = 0\} = \{g\in \Gamma \mid \vert\varphi_{b}(g)\vert = 1\}. \end{align*}$$

It follows from the proof of Proposition 3.1 that $\Lambda $ is a subgroup. Moreover, because of (4.1), $\Lambda $ is finite, hence by torsion-freeness it is the trivial subgroup. Thus, we can conclude by Remark 3.5 that $\varphi _{b}$ is strict, i.e. b has radical growth. ▪

There are many examples of groups and metric spaces satisfying the hypothesis above. This includes trees (see [Reference Bekka, Harpe and Valette2, Sec 2.3]) or even wall spaces (see for instance the comments after [Reference Shalom18, Def 5.1]), and real or complex hyperbolic spaces $\mathbb {H}^{n}_{\mathbb {R}}$ , $\mathbb {H}^{n}_{\mathbb {C}}$ (see [Reference Bekka, Harpe and Valette2, Sec 2.6]). One can also consider product actions on direct products of those spaces. As an illustration, it is shown in [Reference Dreesen8, Subsec 6.4] that Baumslag–Solitar groups $BS(p, q)$ with $p, q> 1$ act geometrically on the product of the corresponding Bass–Serre tree and the real hyperbolic plane. Moreover, we have the following “permanence properties” for the existence of a $1$ -cocycle with radical growth:

1. If $\Gamma _{1}$ and $\Gamma _{2}$ have a $1$ -cocycle with radical growth and $\Lambda $ is a common finite subgroup, then the proof of [Reference Dreesen9, Thm 4.7] shows that the amalgamated free product $\Gamma _{1}\ast _{\Lambda }\Gamma _{2}$ has a $1$ -cocycle with radical growth;

2. If $\Gamma $ is a finitely generated group together with a subgroup $\Lambda $ and a monomorphism $\theta : \Lambda \to \Gamma $ such that $\Lambda \cup \theta (\Lambda )$ generates a finite subgroup of $\Gamma $ , then any $1$ -cocycle with radical growth on $\Gamma $ yields a $1$ -cocycle with radical growth on $ \operatorname {\mathrm {HNN}}(\Gamma , \Lambda , \theta )$ by the proof of [Reference Dreesen9, Thm 4.9].

This gives a wealth of examples, including for instance any group built from Baumslag–Solitar groups, free groups, surface groups and infinite Coxeter groups that one can iterate.