Timeline for The Classifying Space of the Discrete Heisenberg Group

4 events

when toggle format	what		by	license	comment
Apr 18, 2012 at 13:19	vote	accept	Zuriel
Jan 2, 2012 at 9:21	comment	added	Alain Valette		@ Zuriel. The central extension does not split. But you have another short exact sequence $0\rightarrow\mathbb{Z}^2\rightarrow\Gamma\rightarrow\mathbb{Z}\rightarrow 0$ (which of course splits) given as follows: give $\Gamma$ the presentation $\Gamma=<x,y,z\|z=[x,y],[x,z]=[y,z]=1>$. Then the subgroup generated by $z$ and $y$ is normal and isomorphic to $\mathbb{Z}^2$, and the action of $x$ by conjugation on this subgroup is given by the 2-by-2 matrix mentioned in my answer.
Jan 2, 2012 at 2:57	comment	added	Zuriel		@Alain Valette, you have mentioned that $\Gamma$ can be viewed as the semi-direct of $\mathbb{Z}^2$ and $\mathbb{Z}$; did you mean that $n\in\mathbb{Z}$ is mapped to the element $f_n\in Aut(\mathbb{Z}^2)$, such that $f_n(m_1,m_2)=m_1+nm_2$? Then why is $\Gamma$ isomorphic to the semi-direct product? It seems that the central extension $0\rightarrow\mathbb{Z}\rightarrow\Gamma\rightarrow\mathbb{Z}^2\rightarrow0$ does not split; then is there any nature semi-direct structure from this central extension? Thank you!
Jan 1, 2012 at 9:00	history	answered	Alain Valette	CC BY-SA 3.0