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If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?

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  • $\begingroup$ If $G$ is countable, then tehre exists even a proper invariant metric, as can be seen here: math.uni-bielefeld.de/~amanouss/papers/jlms1.pdf $\endgroup$
    – user1688
    May 4, 2013 at 14:48
  • $\begingroup$ Do you mean $\textit{countable}$ group? Otherwise just take the uncountable product of a discrete group. The result is reasonably nice (ie it's a compact topological group). However, the underlying space is not metrizable. $\endgroup$ May 4, 2013 at 18:02
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    $\begingroup$ @Owen Sizemore: there is no compatibility with any given topology asked in the question. $\endgroup$ May 4, 2013 at 18:35
  • $\begingroup$ No compatibility? Then the 0-1 metrics would be an answer. $\endgroup$ May 4, 2013 at 21:26
  • $\begingroup$ @Wlodzimierz: how do you prove that the 0-1 metric is unbounded? :) $\endgroup$
    – YCor
    May 4, 2013 at 21:28

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The answer is no. See Thm 1.2 of http://homepages.math.uic.edu/~rosendal/PapersWebsite/Property(OB)10.pdf. There is a property discussed in the intro of this paper which is equivalent to all left invariant metrics are bounded. It is known that certain large permutation groups have this property.

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    $\begingroup$ This is actually originally due to Bergman (Bull London Math Soc) arxiv.org/abs/math/0401304, Rosendal considers the version for a group endowed with a topology. $\endgroup$
    – YCor
    May 4, 2013 at 20:25
  • $\begingroup$ Thanks Yves. I realized only the intro talks about this but I didn't have a link to Bergman's paper. $\endgroup$ May 4, 2013 at 21:15
  • $\begingroup$ Very interesting answers. Thank you. $\endgroup$
    – H-Hook
    May 5, 2013 at 21:34

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