3 The fixed point characterization of amenability

The goal of this chapter is to give a characterization of amenability via a fixed point property. It will allow us to extend the notion of amenability from groups to topological groups.

3.1 Locally convex spaces and the barycenter map

Let us recall that a locally convex topological vector space is a topological vector space such that the origin admits a basis of convex neighborhoods.

Definition 3.1 (semi-norm) Let \(E\) be a vector space. A semi-norm is a map \(p\colon E\to\mathbf{R_+}\) such that

\(p(\lambda x)=|\lambda|p(x)\) for all \(x\in E\) and \(\lambda\in \mathbf{R}\)
\(p(x+y)\leq p(x)+p(y)\)

Example 3.1 \(\ \)

Any norm is a semi-norm
If \(X\) is compact space and \(E=C(X)\) with the uniform convergence topology then for any \(x\in X\), \(p_x(f)=|f(x)|\), for \(f\in E\), is a continuous semi-norm on \(E\)

Given a vector space \(E\) and a family of semi-norms \(\mathcal{P}\) on \(E\), the topology given by \(\mathcal{P}\) is the coarsest topology on \(E\) that makes each element of \(\mathcal{P}\) continuous. this a vector space topology.

Proposition 3.1 A topological vector space is locally convex if and only if its topology is given by a collection of semi-norms.

Example 3.2 \(\ \)

Let \(E\) be some Banach space and let \(E^*\) be its dual space with the weak-* topology. This topology on \(E^*\) is given by the semi-norms \(\lambda\mapsto |\lambda(x)|\) for \(x\in E\).
A particular case is the following : Let \(K\) be a compact Hausdorff space. Let \(\mathcal{C}(K)\) be the Banach space of continuous real functions on \(K\) with the norm of uniform convergence. Its dual space is the space of Radon measures on \(K\) by Riesz representation theorem. The topology is given by the semi-norms \(\mu\mapsto|\mu(f)|=\left|\int_Kf(x)d\mu(x)\right|\) for \(f\in\mathcal{K}\).

Proposition 3.2 (Barycenter map) Let \(K\) be a convex compact subset of some locally convex topological vector space \(E\). For any probability measure \(\mu\) on \(K\), there is a unique barycenter \(\textrm{bar}(\mu)\in K\) such that for continuous affine \(f\colon K\to \mathbf{R}\) \(f(\textrm{bar}(\mu))=\mu(f)\).

Morever if \(g\) is an affine continuous transformation of \(K\) then \(\mathrm{bar}(g_*\mu)=g(\mathrm{bar}(\mu))\).

Proof. Uniqueness follows from Hahn-Banach separation theorem: for \(x\neq y\), one can find a linear continuous map \(f\) such that \(f(x)\neq f(y)\). So if \(x,y\) are two barycenters then \(f(x)=\mu(f)=f(y)\) and we have a contradiction.

For the existence, Since \(\{x\in K,\ f(x)=\mu(f)\}\) is compact, by the finite intersection property, it suffices to prove that for any \(f_1,\dots,f_n\) continuous affine maps on \(E\), there is \(x\in K\), such that \(f_i(x)=\mu(f_i)\) for all \(i\). So let us consider the map \[\alpha\colon x\mapsto (f_1(x),\dots,f_n(x)).\]

The space \(\alpha(K)\) is compact convex subspace of \(\mathbf{R}^n\). Let us define \(p=(\mu(f_1),\dots,\mu(f_n))\). We claim that \(p\in \alpha(K)\). If \(\mu\) is a convex combination of Dirac masses \(\mu=\sum_{j=1}^kc_j\delta_{x_j}\) with \(x_j\in K\) and \(c_j\geq0\) with \(\sum_{j=1}^nc_j=1\) then \(p=(f_1(\sum_{j=1}^nc_jx_j),\dots, f_n(\sum_{j=1}^nc_jx_j))\in\alpha(K)\). Now \(\mu\mapsto p\) is continuous and convex combinations of Dirac masses are dense in \(\mathrm{Prob}(K)\) for the weak-* topology. Since \(\alpha(K)\) is closed then \(p\in\alpha(K)\) for any \(\mu\in \mathrm{Prob}(K)\). By construction \(p\in\cap_{i=1}^n\{x\in K,\ f_i(x)=\mu(f_i)\}\), and thus the finite intersection property holds. Thus there is a unique point \(\mathrm{bar}(\mu)\) in the intersection over all continuous affine functions \(f\) of \(\{x\in K,\ f(x)=\mu(f)\}\).

Finally, let \(g\) be some continuous affine transformations of \(K\) then we have for any continuous affine function \(f\) on \(K\):

\[f(\mathrm{bar}(g_*\mu))=g_*\mu(f)=\mu(f\circ g)=(f\circ g)\mathrm{bar}_\mu=f(g\mathrm{bar}_\mu)\] and thus \(\mathrm{bar}(g_*\mu)=g\mathrm{bar}_\mu\).

Let \(K\) be a convex subset of vector space \(E\). Assume a group \(G\) is acting on K. The action is said to be affine if for any \(g\in G\), \(x,y\in K\) and \(t\in[0,1]\), \(g(tx+(1-t)y)=tg(x)+(1-t)g(y)\).

Example 3.3 If \(G\) acts by affine transformations on \(E\) and \(K\) is \(G\)-invariant then the action on \(K\) is affine. This situation is not the general one.

3.2 Day’s fixed point theorem

Theorem 3.1 (Day's fixed point theorem) Let \(G\) be a group. Then \(G\) is amenable if and only if for any affine action on compact convex subset of a locally convex topological vector space, there is a fixed point.

Proof. Let’s assume \(G\) is an amenable group acting affinely on some compact convex set \(K\). Let \(m\) be a left invariant mean on \(G\). For \(f\in\mathcal{C}(K)\) and \(x\in K\), the map \(\theta_f\colon g\mapsto f(gx)\) belongs to \(\ell^\infty(G)\). So let us define

\[\mu(f)=m(\theta_f).\]

One checks that \(\mu\) is a positive linear maps on \(\mathcal{C}(K)\) such that \(\mu(\mathbf{1}_K)=1\), so \(\mu\) is a probability measure on \(K\).

Moreover, \(h_*\mu(f)=\mu(f\circ h)=m(\theta_{f\circ h})=\mu(f)\) since \(m\) is left invariant.

Conversely, the space of means on \(G\) is compact convex subspace of the dual space of \(\ell^\infty(G)\) with the weak-* topology that is invariant under the action of \(G\). Since there is a fixed point, there is an invariant mean.

Corollary 3.1 Any amenable group \(G\) acting by homeomorphisms on compact Hausdorff space \(X\) has an invariant probability measure.

Proof. The space of probability measures \(\mathrm{Prob}(X)\) is a convex compact subspace of the dual space of \(\mathcal{C}(X)\) (for the weak-* topology) which is invariant the affine action of \(G\) on the space of Radon measures on \(X\). Thus there is a fixed point.

3.3 Amenable topological group

Definition 3.2 A topological group \(G\) is amenable if for any continuous affine actions on a convex compact topological vector space, there is a fixed point.

Remark. Day’s fixed point theorem tells us that a group is amenable if and only if it amenable for the discrete topology.

Remark. If \(\tau_1\) and \(\tau_2\) are two group topologies on \(G\), \(\tau_1\) is finer than \(\tau_2\) then if \(G\) is amenable for \(\tau_1\) then it is also amenable for \(\tau_2\).

In particular, if a group is amenable as an abstract group then it is amenable with respect to any topology. For example, finite groups, Abelian or more generally solvable groups are amenable with respect to any group topology.

Theorem 3.2 (Kakutani fixed point theorem) Let \(E\) be a locally convex topological vector space. For any nonempty compact convex set \(C\) in \(E\), any continuous function affine \(f\colon C\to C\) has a fixed point.

Proof. Let \(x\in C\). Define \(x_n=\frac{1}{n+1}\sum_{k=0}^nf^k(x)\). Let \(y\) be some accumulation point of \((x_n)\). If \(f(y)\neq y\) then by Hahn-Banach theorem, there is \(\varphi\in E^*\) such that \(\varphi(y)\neq\varphi(f(y))\) but \(|\varphi(x_n)-\varphi(f(x_n))|\leq\left|\frac{\varphi(x)-\varphi(f^{n+1}(x))}{n+1}\right|\). Since \(\varphi\) is bounded on \(C\), this last term converges to 0 and we have a contradiction.

Theorem 3.3 (Kakutani fixed point theorem) Let \(E\) be a locally convex topological vector space. Let \(C\) be a nonempty compact convex set and \(S\) a family of commuting continuous affine functions \(f\colon C\to C\). Then the elements of \(S\) have a common fixed point.

Proof. We know that for any \(f\in S\), \(\mathrm{Fix}(f)=\{x\in C,\ f(x)=x\}\) is non empty. We want to prove that

\[\cap_{f\in S}\mathrm{Fix}(f)\neq\emptyset.\]

Thanks to the finite intersection, it suffices to prove that for any \(f_1, \dots,f_n\in S\), \(\cap_{i=1}^n\mathrm{Fix}(f_i)\neq\emptyset.\)

We prove the emptyness of this finite intersection by induction on \(n\). The base case (\(n=1\)) is the Kakutani fixed point theorem. Assume, we have \(\cap_{i=1}^{n-1}\mathrm{Fix}(f_i)\neq\emptyset.\) Since \(f_n\) commute with each \(f_i\), \(\mathrm{Fix}(f_i)\) is \(f_n\)-invariant and we can apply Kakutani theorem to \(C=\cap_{i=1}^{n-1}\mathrm{Fix}(f_i)\) and \(f_n\), that is \(f_n\) has a fixed point in \(C=\cap_{i=1}^{n-1}\mathrm{Fix}(f_i)\) and thus \(\cap_{i=1}^n\mathrm{Fix}(f_i)\neq\emptyset.\)

Corollary 3.2 Abelian groups are amenable for the discrete topology.

Theorem 3.4 Compact groups are amenable.

Proof. We prove that any compact group has an invariant probability measure (a particular case of the existence of Haar measures for locally compact groups).

Let \(G\) be a compact group. It suffices to deal with the case where \(G\) is infinite. Let \(\mu\) be some diffuse probability measure on \(G\) with full support. Let \(\lambda\) be some \(\mu\)-stationnary probability measure, i.e. \(\mu\ast \lambda=\lambda\). Such measure exists thanks to Tychonoff fixed theorem.

For \(f\in\mathcal{C}(G)\), let us define

\[\varphi(g)=\int_G f(gh)d\lambda(h).\] The function \(\varphi\) is continuous and thus has a maximum at some \(g_0\). Up to replace \(f\) by \(f'\colon x\mapsto f(g_0x)\), we may assume that \(\varphi\) achieves its maximum at \(e\). By stationnarity

\[\int_G\varphi(g)d\mu(g)=\iint f(gh)d\mu(g)d\lambda(h)=\int f(h)d\lambda(h)=\varphi(e).\]

So, \(\mu\)-a.s. \(\varphi(g)=\varphi(e)\). Since \(\mu\) has full support and \(\varphi\) is continuous, \(\varphi(g)=\varphi(e)\) for all \(g\in G\). That \(\varphi\) is constant and \(\lambda\) is invariant.

Now, let \(K\) be some convex compact space in some locally convex space \(E\) and assume that \(G\) by continuous affine maps on \(K\). We want to prove there is a fixed point in \(K\). Let \(x\in K\). We define the orbit map \(f\colon G\to K\) by \(f(g)=g\cdot x\). If \(\lambda\) is the invariant probability on \(G\) found above then \(f_*\lambda\) is an invariant probability on \(K\). Its barycenter \(\mathrm{bar}(f_*\lambda)\) is a \(G\)-fixed point.

Remark. The topological group \(\mathrm{SO}(3)\) is not amenable with the discrete topology because of the paradoxical decompositions we have seen in the first chapter, whereas it is amenable with its usual topology since it is compact.

Definition 3.3 Let \(G\) a topological group. A function \(f\colon G\to \mathbf{R}\) is left uniformly continuous if for any \(\varepsilon>0\), there is an identity neighborhood \(U\) such that for \(g,h\in G\), \(gh^{-1}\in U\implies |f(g)-f(h)|<\varepsilon\).

We denote by \(\mathrm{UCB}(G)\) the space of left uniformly continuous bounded functions on \(G\). With the \(\sup\) norm, this is a Banach space. Moreover, for \(f\in\ell^\infty(G)\), the map \(g\mapsto g\cdot f=f\circ g^{-1}\) is continuous if and \(f\) is left uniformly continuous and the \(G\) acts on \(\mathrm{UCB}(G)\) by left translations.

Definition 3.4 Let \(G\) be a topological group. A mean on \(\mathrm{UCB}(G)\) is a linear map \(m\colon \mathrm{UCB}(G)\to\mathbf{R}\) such that

\(m(\mathbf{1}_G)=1\)
if \(f\geq0\) then \(m(f)\geq0\).

Proposition 3.3 Let \(G\) be a topological group. The following are equivalent

\(G\) is amenable
There is a \(G\)-invariant mean on \(\mathrm{UCB}(G)\).

Proof. The space of means on \(\mathrm{UCB}(G)\) is compact convex subspace in the dual space \(\mathrm{UCB}(G)^*\) and \(G\) acts continuously by affine maps. So if \(G\) is amenable, there is an invariant mean on \(\mathrm{UCB}(G)\).

Conversely, assume that there is a \(G\)-invariant mean \(\lambda\) on \(\mathrm{UCB}(G)\). Let \(K\) be some convex compact space in some locally convex space \(E\) and assume that \(G\) by continuous affine maps on \(K\). We want to prove there is a fixed point in \(K\). Let \(x\in K\). We define the orbit map \(f\colon G\to K\) by \(f(g)=g\cdot x\).

Let \(\varphi\in\mathcal{C}(K)\). This map is uniformly continuous and since the action of \(G\) on \(K\) is continuous, for any \(\varepsilon>0\), there is an open neighborhood \(U\) of the identity in \(G\) such that \(|\varphi(ux)-\varphi(x)|<\varepsilon\). So \(\varphi\circ f\in\mathrm{UCB}(G)\) and we can define \(\lambda(\varphi\circ f)\). The map \(\varphi\mapsto \lambda(\varphi\circ f)\) defines a probability measure on \(K\) that is \(G\)-invariant. Its barycenter is a \(G\)-fixed point.

Proposition 3.4 Let We have the following stability properties

If \(h\colon G\to H\) is a continuous surjective homomorphism between topological groups \(G,H\) then if \(G\) is amenable then \(H\) is amenable.
If \(G\) has a dense amenable subgroup (for the induced topology) then \(G\) is amenable
If \(G\) contains a dense directed unions of amenable subgroups then \(G\) is amenable.
The topological group \(G\) is amenable if the closure of any finitely generated subgroup is amenable.
If \(H\) is an open subgroup of \(G\) and \(G\) is amenable then \(H\) is amenable.
Let \(N\) be a normal subgroup of \(G\). Then \(G\) is amenable if \(N\) and \(G/N\) are amenable. Moreover, if \(N\) is open this is an equivalence.

Proof.

Consider an affine continuous action of \(H\) on a compact convex space \(C\): \(\alpha\colon H\times C\to C\). Consider the action \(\beta\colon G\times C\to C\) given by \(\beta(g,x)=\alpha(h(g),x)\). This is a continuous affine action of \(G\) on \(C\), so there is a \(G\)-fixed point and by construction, this point is also \(H\)-fixed.
Let \(H\) be a dense amenable subgroup of \(G\). Let \(\alpha\colon G\times C\to C\) be a continuous affine action of \(G\) on a compact convex space \(C\). By amenability of \(H\), there is \(x\) in \(C\) such that \(\alpha(h,x)=x\) for all \(h\in H\). By denseness of \(H\) and continuity of \(\alpha\), \(\alpha(g,x)=x\) for all \(g\in G\).
By the previous point, it suffices to prove that a directed union of amenable groups is amenable. Let us recall that a directed set is a poset \((I,\leq)\) such that for any \(i,j\in I\), there is \(k\in I\) such that \(i,j\leq k\). A directed union \(\cup_{i\in I}X_i\) is a union index by a directed set \(I\) such that \(X_i\subset X_j\) for \(i\leq j\). Let \(G\) be some topological group and \((G_i)_{i\in I}\) be a collection of subgroups of \(G\) such that \(G_i\subset G_{j}\) for \(i\leq j\). Let \(C\) be a compact convex \(G\)-space. We want to prove that \(\cup_{i\in I}G_i\) has a fixed point i.e. \(\cap_{i\in I}\mathrm{Fix}(G_i)\neq\emptyset\). By the finite intersection property, it suffices to prove that \(\cap_{i\in F}\mathrm{Fix}(G_i)\neq\emptyset\) where \(F\subset I\) is finite. This intersection containing \(\mathrm{Fix}(G_k)\) where \(k\geq i\) for all \(i\in F\), we have the result by amenability of \(G_k\).
The union of all finitely generated groups of \(G\) is directed union and we can apply the previous result.
Let \(H\) be an open subgroup. So there is \(U\) open identity neighborhood such that \(U\subset H\). In particular, for any \(h\in H\), \(hU\subset H\). For each \(i\in H\backslash G\) we choose \(g_i\in G\) such that \(Hg_i=i\) and \(g_H=e\). For \(f\in\mathrm{UCB}(H)\), we define \(\overline{f}\colon G\to\mathbf{R}\) by \(\overline{f}(g)=f(gg^{-1}_i)\) for \(g\in i\). So \(||\overline{f}||_\infty=||f||_{\infty}\) and for \(\varepsilon >0\), there is \(V\subset U\) such that \(\forall g,h\in H\), \(gh^{-1}\in U \implies |f(g)-f(h)|<\varepsilon\). Now for \(g,h\in G\) such that \(gh^{-1}\in U\), we have \((gg_i^{-1})(hg_i^{-1})\in U\) and \((gg_i^{-1}),(hg_i^{-1})\in H\), so \(|f(gg_i^{-1})-f(hg_i^{-1}|<\varepsilon\), i.e. \(|\overline{f}(g)-\overline{f}(h)|<\varepsilon\). So \(\overline{f}\in \mathrm{UCB}(G)\). Observe that \(\overline{\mathbf{1}_H}=\mathbf{1}_G\) and \(\overline{f+\lambda g}=\overline{f}+\lambda\overline{g}\), that is \(f\mapsto \overline{f}\) is linear. Moreover for \(h\in H\), \(g\in G\) and \(f\in\mathrm{UCB}(H)\), we have

\[\overline{h\cdot f}(g)=h\cdot f(gg_i^{-1})=f(h^{-1}gg_i^{-1})=\overline{f}(h^{-1}g)=h\cdot\overline{f}(g).\] So, \(\overline{h\cdot f}=h\cdot\overline{f}\). Let \(\mu\) be a \(G\)-invariant mean on \(\mathrm{UCB}(G)\). We define \(\nu(f)=\mu(\overline{f})\) for \(f\in \mathrm{UCB}(H)\). This is a non-negative linear map on \(\mathrm{UCB}(H)\) with \(\nu(\mathbf{1}_H)=1\), so it is a mean. The \(H\)-invariance follows from the following computation:

\[h\nu(f)=\nu(h^{-1}f)=\mu(\overline{h^{-1}f})=\mu(h^{-1}\overline{f})=\mu(\overline{f})=\nu(f).\] 6. Let \(C\) be a compact convex \(G\)-space. Since \(N\) is amenable, it has a fixed point in \(C\), i.e. \(\mathrm{Fix}(N)\) is a non-empty compact convex subspace. Since \(N\) is normal, \(\mathrm{Fix}(N)\) is \(G\)-invariant and since \(N\) acts trivially, this gives a \(G/N\) compact convex space and thus by amenability of \(G/N\), there is a \(G/N\)-fixed point. This point is also fixed for the original \(G\) action. In conclusion \(G\) is amenable. The converse direction follows from points 1 and 5 above.

Corollary 3.3 We can deduce the following amenability results

The group of all permutation \(S_\infty\) of a countable set with the pointwise convergence is amenable
The group of all unitary operators of a Hilbert space is amenable for the weak operator topology.

Corollary 3.4 Let \(G\) be an amenable discrete group. Then any subgroup \(H\leq G\) is amenable as well.

Proof. Since \(G\) is discrete, any subgroup is open and we can use the above proposition.

Corollary 3.5 Any solvable group is amenable for any group topology.

Proof. A group \(G\) is solvable if the derived series \(G_n\) eventually satisfies \(G_{n}=\{e\}\) where \(G_0=G\) and \(G_{n+1}=[G_n,G_n]\). The result is proved by induction on the solvability order (the least \(n\) such that \(G_{n}=\{e\}\)). The base case (\(n=0\)) is obvious. The induction step follows from the fact that \(G_n/G_{n+1}\) is Abelian.

Example 3.4 The subgroup of upper triangular matrices is amenable

3.4 Locally compact amenable groups

Locally compact groups are remarkable among topological groups. Namely, such groups are characterized by the existence of Haar measures.

Theorem 3.5 Let \(G\) be a locally compact group then there is left invariant positive Radon measure on \(G\). It is unique up to multiplication by some positive number.

Conversely, a topological group with a left invariant Borel measure that is locally finite and inner regular is locally compact.

The main point about locally compact groups for us will be the following point.

Theorem 3.6 Let \(G\) be a locally compact group and \(H\) be a closed subgroup. If \(G\) is amenable then \(H\) is amenable.

Remark. If \(G\) is not locally compact, one can find closed subgroup that are not amenable. For example, by letting \(\mathbf{F}_2\) acting on itself by left multiplications, we can realize \(\mathbf{F}_2\) as a discrete subgroup and thus as a closed subgroup of \(S_\infty\) but \(\mathbf{F}_2\) is not amenable!

We will prove the result under two supplementary hypotheses : The quotient space is paracompact (from any open covering one can extract a locally finite subcovering) and the existence of local cross sections.

Definition 3.5 Let \(G\) be a topological group and \(H\) a closed subgroup. We denote by \(\pi\colon G\to G/H\) the quotient and endow \(G/H\) with the quotient topology. For \(g\in G\), a local cross section at \(\pi(g)\) is a continuous map \(s\colon U\to G\) where \(U\) is an open neighborhood of \(\pi(g)\) and for any \(x\in \pi(U)\), \(\pi(s(x))=x\).

Local cross sections exist under very general hypotheses. Actually if \(G\) is separable and metrizable locally compact group then local cross sections always exist. Let us prove this existence for Lie groups.

Proposition 3.5 Let \(G\) be a Lie group and \(H\) be a closed subgroup then local cross sections exists.

Proof. We first prove the existence of local cross section at \(e\).

Since \(H\) a closed subgroup then \(H\) is a Lie subgroup. Let \(\mathfrak{g}\) be the Lie algebra of \(G\) and \(\mathfrak{h}\) be the one of \(H\). let \(\mathfrak{h}'\) be a supplementary linear subspace of \(\mathfrak{h}\), i.e. \(\mathfrak{g}=\mathfrak{h}'\oplus\mathfrak{h}\). Let us define

\(\varphi(v_1+v_2)=\exp(v_1)\exp(v_2)\) for \(v_1\in\mathfrak{h}'\) and \(v_2\in\mathfrak{h}\). So \(\varphi\) is a map from \(\mathfrak{g}\) to \(G\) and thanks to the Baker-Campbell-Hausdorff its differential at 0 is the identity. So there is \(U_0\) and \(U\) such that \(\varphi\) induces a diffeomorphism beteween \(U_0\) and \(U\). Now we define \(s\colon \pi(U)\to U\) via the formula

\(s(\pi(g))=\exp(v_1)\) where \(v_1\) is such that \(g=\exp(v_1)\exp(v_2)\). This is a continuous map and \(\pi\circ s=\mathrm{id}.\)

Now if \(s\) is a local cross section at \(e\) then for any \(g\in G\), \(gs\) is a local cross section at \(g.\)

Proof. So let’s assume that \(G/H\) is paracompact and there are local cross sections.

In particular, there is open covering \((U_i)\) of \(G/H\) that is locally finite and local sections \(s_i\colon U_i\to G\). Let \((\lambda_i)\) be a partition of unity relative to the covering \(\pi^{-1}(U_i)\). Let \(f\in \mathrm{UCB}(H)\). Define \(f_i\) on \(\pi^{-1}(U_i)\) by \(f_i(g)=f(gs_i(\pi(g))^{-1})\).

Let \[f'(g)=\sum_{i}f_i(g)\lambda_i(\pi(g))\].

We define \(m'(f)=m(f')\) where \(m\) is a left invariant mean on \(\mathrm{UCB}(G)\). This is a left invariant on \(\mathrm{UCB}(H)\).

Corollary 3.6 The group \(\mathrm{SL}_n(\mathbf{R})\) is not amenable for \(n\geq 2\).

Proof. The group \(F\) generated by matrices \[\begin{bmatrix}1&2&&&\\0&1&&&\\&&0&&\\&&&\ddots&\\&&&&0\end{bmatrix}\] and \[\begin{bmatrix}1&0&&&\\2&1&&&\\&&0&&\\&&&\ddots&\\&&&&0\end{bmatrix}\] is closed since it is subgroup of \(\mathrm{SL}_n(\mathbf{Z})\) which is discrete in \(\mathrm{SL}_n(\mathbf{R})\). We have seen that \(F\) is a free group on 2 generators and thus is not amenable. So \(\mathrm{SL}_n(\mathbf{R})\) is not amenable.

3.5 Exercices

Exercise 3.1 Prove that the group of transformations that preserves angles in \(\mathbf{R}^n\) is amenable for the compact open topology.

Exercise 3.2 Let \(\mathcal{H}\) be a Hilbert space.

Prove that any isometry of \(\mathcal{H}\) is an affine transformation.
Prove that the linear part of an isometry is an orthogonal transformation.
Prove the isometry group \(\mathrm{Isom}(\mathcal{H})\) is the semi-direct product \((\mathcal{H},+)\rtimes O(\mathcal{H})\) where \(O(\mathcal{H})\) is the orthogonal group of \(\mathcal{H}\).
Let us endow \(\mathrm{Isom}(\mathcal{H})\) with the pointwise convergence topology (i.e. the coarsest topology on \(\mathrm{Isom}(\mathcal{H})\) such that all maps \((g,h)\mapsto d(g(x),h(x))\) and \((g,h)\mapsto d(g^{-1}(x),h^{-1}(x))\) where \(x\in\mathcal{H}\) are continuous). Prove this a group topology.
Prove that \(\mathrm{Isom}(\mathcal{H})\) is amenable.

Exercise 3.3 Let \(G\) be a topological group.

Prove there is a unique maximal (for inclusion) amenable normal subgroup of \(G\). It is called the amenable radical of the group.
Prove that the amenable radical is a closed subgroup.

Exercise 3.4 Let \(G\) be a topological group and \(H\) a normal closed subgroup such that \(G/H\) is compact for the quotient topology. Prove that if \(H\) is amenable then \(G\) is amenable and if \(G\) is locally compact this is an equivalence. Is it still if one removes the hypothesis that \(H\) is normal?

Exercise 3.5 Prove that any product (finite or not) of amenable groups is amenable for the product topology.