Is it possible for a real semisimple Lie group to have infinitely many connected components? If so, what is an example?
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2026-04-09 14:10:10.1775743810
Real semisimple Lie group with infinitely many connected components?
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Yes, $$\mathrm{SL}(2,\mathbb{R})\times\mathbb{Z}.$$ (Or any real semisimple group times a countably infinite discrete group.) The point is that $$\mathrm{Lie}(G\times H)=\mathrm{Lie}(G)\times\mathrm{Lie}(H)$$ and $\mathrm{Lie}(H)=\{0\}$ for a discrete group $H$.