almost connected topological group


Group Theory





A locally compact topological group GG is called almost connected if the underlying topological space of the quotient topological group G/G 0G/G_0 (of GG by the connected component of the neutral element, also called the identity component) is compact.

See for instance (Hofmann-Morris, def. 4.24). We remark that since the identity component G 0G_0 is closed, the identity in G/G 0G/G_0 is a closed point. It follows that G/G 0G/G_0 is T 1T_1 and therefore, because it is a uniform space, T 312T_{3 \frac1{2}} (a Tychonoff space; see uniform space for details). In particular, G/G 0G/G_0 is compact Hausdorff.


Every compact and every connected topological group is almost connected.

Also every quotient of an almost connected group is almost connected.


Textbooks with relevant material include

  • M. Stroppel, Locally compact groups, European Math. Soc., (2006)

  • Karl Hofmann Sidney Morris, The Lie theory of connected pro-Lie groups, Tracts in Mathematics 2, European Mathematical Society, (2000)

Original articles include

  • Chabert, Echterhoff, Nest, The Connes-Kasparov conjecture for almost connected groups and for linear pp-adic groups (pdf)

Revised on February 3, 2012 22:02:40 by Todd Trimble (