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Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the action of $G$ becomes an isometric action? Conversely, given a metric on $X$, is there a nontrivial action of $G$ on $X$ that preserves the metric?

I am looking for the most general necessary and sufficient conditions and any possible obstructions. For the first question, the answer is obviously positive when $G$ is compact: one chooses a metric $d$ on $X$ and simply does an "averaging process along the orbits" by defining $$\rho(x,y)= \int_{G} d(g.x,g.yd(g^{-1}.x,g^{-1}.y) dg.$$ I suspect that a similar idea would work more generally using a "cut-off" function on $X$ when the action of $G$ on $X$ is proper. Any connections to amenability (of the group or the group action) would also be interesting.

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Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$ X$, compatible with the topology, for which the action of$G$becomes an isometric action? Conversely, given a metric on$X$, is there a nontrivial action of$G$on$X$that preserves the metric? I am looking for the most general necessary and sufficient conditions and any possible obstructions. For the first question, the answer is obviously positive when$G$is compact: one chooses a metric$d$on$X$and simply does an "averaging process along the orbits" by defining $$\rho(x,y)= \int_{G} d(g.x,g.y) dg.$$ I suspect that a similar idea would work more generally using a "cut-off" function on$X$when the action of$G$on$X\$ is proper. Any connections to amenability (of the group or the group action) would also be interesting.

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# When do isemtricisometric actions exist?

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