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Let $G$ be any locally compact group and $H$ be a compact group.

We know that a map $F: G \rightarrow G$ is called affine if there exists some $\alpha \in G$ and an automorphism $\Lambda:G\rightarrow G$ such that $F=\alpha\Lambda$.

A continuous action of $H$ on $G$ is a continuous map $\pi: H\times G \rightarrow G$ given by $\pi(h, g) = h.g$ such that $e.g =g$ for all $g\in G$ and $(hk).g= h. (k.g)$ for all $h, k \in H$, $g\in G$.

Hence a continuous action $\pi$ of $H$ on $G$ is called a continuous affine action if for each $h\in H$, the map $g\mapsto (h.g): G \rightarrow G$ is affine.

My Question: Could you please give me an example of such a continuous affine action where atleast one of the maps $g\mapsto (h.g): G \rightarrow G$ is (of course) affine, but not an automorphism?

Thank you in advance, for your help.

Alp Uzman
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Carl Butcher
  • 330

2 Answers2

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Let $G=(\mathbb{R},+)$ and $H=C_2=\langle t\rangle$. Then we may define: $$t.x=1-x=\Lambda(x)+1,$$ for all $x\in \mathbb{R}$, where $\Lambda(x)=-x$ for all $x\in \mathbb{R}$.

Alternatively, you could take $G=H=S^1\subset \mathbb{C}^\times$. For $g\in G$ and $h\in H$ we can define $h.g=hg=h\Lambda(g)$, where $\Lambda$ is the identity.

For a third example, consider $G=SU(2)\times S^1$ and $H=S^1\subset \mathbb{C}^\times$. Let $\Lambda_h\colon G\to G$ denote conjugation by:$$\left(\begin{array}{cc}h&0\\0&\bar{h}\end{array}\right)$$ on the factor $SU(2)$, and the identity on the factor $S^1$.

Then we may define $$h.g=(1,h)\Lambda_h(g).$$

tkf
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As a slight generalization to the second example of tkf, for any locally compact (Hausdorff) group $H$ one can consider the group $\operatorname{Aff}(H)=H\rtimes \operatorname{Aut}(H)$ of affine automorphisms of it. By definition, any element of $\operatorname{Aff}(H)$ is of the form $(g,\phi)$, where $g\in H$ and $\phi:H\to H$ is a topological group automorphism. A $(g,\phi)\in \operatorname{Aff}(H)$ acts on $H$ like so:

$$(g,\phi):H\to H, x\mapsto g\phi(x)=l_g\circ \phi(x),$$

where $l_g$ is the left translation by $g$. Note that the group structure of $\operatorname{Aff}(H)$ is given by: $(g,\phi)(h,\psi)=(g\phi(h), \phi\circ \psi)$. In other words, what you call "affine maps" of $H$ assemble into a group that acts via affine maps on $H$.

In particular, if $H$ is a compact group, $G=\operatorname{Aff}(H)$ is a locally compact group, and $G\curvearrowright H$ is an action by affine automorphisms where most elements are not automorphisms (i.e. most elements don't fix the identity element $e_H$ of $H$). One can also take subgroups of $\operatorname{Aff}(H)$ (that has non-trivial translation part) to get further examples (in fact, any (faithful) affine action on $H$ is of this latter form, by definition).

For example if $H=\mathbb{T}^d$ ($d\in\mathbb{Z}_{\geq1}$), then $G=\operatorname{Aff}(\mathbb{T}^d)= \mathbb{T}^d\rtimes \operatorname{Aut}(\mathbb{T}^d)\cong \mathbb{T}^d\rtimes \operatorname{GL}(d,\mathbb{Z})$ (e.g. by https://math.stackexchange.com/a/4180086/169085). Taking the subgroup $\mathbb{T}^d\rtimes 1\leq G$, for $d=1$ we recover tkf's second example.

Above I skipped one detail: since the action $\operatorname{Aff}(H)\curvearrowright H$ is supposed to be topological, we need to specify a topology on $\operatorname{Aff}(H)$. $\operatorname{Aff}(H)$ is a semidirect product, so as a set it is a cartesian product. Accordingly we may define the topology of $\operatorname{Aff}(H)$ as the product topology (see e.g. How to construct a topological group from a semidirect product of two subgroups). So specifying the topology of $\operatorname{Aff}(H)$ reduces to specifying a topology on $\operatorname{Aut}(H)$. The standard way of doing this is to consider $\operatorname{Aut}(H)$ as a subgroup of the group $\operatorname{Homeo}(H)$ of homeomorphisms of $H$, for which there is a variety of natural topologies, e.g. the compact-open topology (= topology of convergence on compact subsets) (see https://mathoverflow.net/q/413361/66883). In the example of the affine actions on the torus there are more options as there is a differentiable structure, and one can consider $\operatorname{Aut}(\mathbb{T}^d)$ as a subgroup of the group $\operatorname{Diff}^r(\mathbb{T}^d)$ of $C^r$ diffeomorphisms of the torus (or the group of volume preserving diffeomorphisms, ...). In any event, the continuity of the affine action will be pointwise continuity, which is straightforward to guarantee.

Alp Uzman
  • 10,742