Let $U$ be a unipotent upper triangluar group over a local field $K$ of characteristic zero. Can we guarantee that there is a right translation invariant metric on $U$ such that any ball of finite radius is relatively compact?
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Let $G$ be a locally compact group. If $G$ is compactly generated, then word length with respect to a compact generating subset defines an invariant metric which is proper (i.e. closed balls are compact). The problem here is that a unipotent group $U$ is usually not compactly generated (if $K$ is non-archimedean). But it can be naturally embedded as a closed subgroup in a compactly generated group, e.g. the subgroup $B$ of all triangular matrices. So take the word metric in $B$ and restrict to $U$.
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$\begingroup$ Sorry, I'm a bit confused: The word metric as I understand it induces the discrete topology, but its balls are compact with respect to the original topology, right? So what do you do if you want a proper metric inducing the original topology? Is there an easy trick using convolutions? $\endgroup$ Commented May 3, 2011 at 17:31
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1$\begingroup$ Indeed in the OP I interpreted "relative compact" as wrt the original compact. On the other hand I may direct you to a paper by Haagerup and Przybyszewska arxiv.org/pdf/math/0606794 I copy the beginning of the abstract: "In this article it is proved, that every locally compact, second countable group has a left invariant metric $d$, which generates the topology on $G$, and which is proper, ie. every closed $d$-bounded set in $G$ is compact." $\endgroup$ Commented May 3, 2011 at 21:07
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$\begingroup$ Thank you very much! I need a little time to think about the construction a bit, but it looks quite nice. $\endgroup$ Commented May 3, 2011 at 21:30