Tag Archives: free

Discrete groups and FPD actions

Joey’s question about discreteness and FPD actions caught me a little unprepared this morning, and I may not have answered as clearly as I should.  If \(G\) is a topological group, then the notion of action of \(G\) on \(X\) needs to be modified to require that \(g \mapsto g\cdot x\) should be a continuous function of \(g\), for each fixed \(x\).  (Note that the continuity in \(x\) of such an expression follows from the isometry condition.)  Now if a metrizable (or Hausdorff if you prefer) topological group \(G\) acts on a metric space via an FPD action, then the topology of \(G\) must be discrete. (If not, consider a sequence \( \{g_n\}\) of non-identity elements of \(G\) that converges to the identity, and ask what happens to \(g_n\cdot x\).)  Thus discreteness of \(G\) is actually an automatic consequence of the other conditions; for which reason, I didn’t mention it at all.

I see that the in-class exercise for this time got pushed to the next lecture.  So, whoever it was who was due to do today’s in-class assignment, try writing out and posting  the details of the above argument instead (maybe in a comment to this post).

John