Complete metrics in locally compact topological groups

Hello,

I am trying to show that every metrizable locally compact topological group admits a complete metric generating the topology of the group

• Could you please give some background or motivation (why do you want to know? why do you think it's true? for which examples do you know it's true? etc) – Yemon Choi Aug 15 '11 at 20:41
• suggestions: Choose one side, say left. Show that the left uniformity is complete. Show that a left-invariant metric exists. Relate these two. – Gerald Edgar Aug 15 '11 at 21:24
• If you make the completion of the metric space (taking the Cauchy sequences) this is coherent by algebraic group operations – Buschi Sergio Aug 15 '11 at 21:56

Every second countable, locally compact group admits a metric which is left-invariant, generates the topology, and is proper (i.e. closed balls are compact). See Theorem 4.5 in http://arxiv.org/pdf/math/0606794

Such a metric is clearly complete.

• However, the solution for the homework-type problem proposed was known long before 2006. – Gerald Edgar Aug 16 '11 at 12:06
• @Gerald: Indeed, Yves Cornulier just pointed out to me this paper: Struble, Raimond A. Metrics in locally compact groups. Compositio Mathematica, 28 no. 3 (1974), p. 217-222 numdam.org/numdam-bin/fitem?id=CM_1974__28_3_217_0 – Alain Valette Aug 16 '11 at 14:36
• A related result from 1951: "Invariant Metrics in Groups" by V.L. Klee Jr., in which it is shown that every Abelian, completely metrizable group admits a complete invariant metric. – Alex M. Jan 25 '18 at 15:46
• @AlainValette I cannot see the link anymore...to the numdam paper, do you know where it has traveled? Merci en avance :) – AIM_BLB Jan 22 at 14:37