Let $V$ be an $n$-dimensional real vector space, $\Lambda\subseteq V$ a lattice, and $K$ a subgroup of $Aut_{\mathbb{Z}}(\Lambda)\cong GL(n,\mathbb{Z})$. Let also $\sigma \in Z^1(K,V/\Lambda)$, $\sigma:K\to V/\Lambda$, be a cocycle, and $s:K\to V$ a lift of $\sigma$. Define

$K \ltimes^{\sigma} \Lambda:=\{ (k,\lambda + s(k))\in K\times V\; | \; k\in K,\; \lambda \in \Lambda \}\subseteq K \times V.$

Does the above construction have a name (maybe in more general form)? Is there a standard notation for it?

Remark: given a finite $K\subset GL(n,\mathbb{Z})$, it is used to construct the crystallographic group in the arithmetic crystal class $(K,\mathbb{Z}^n)$ corresponding to the cocycle $\sigma$.


This group fits into a short exact sequence $1 \to \Lambda \to G \to K \to 1$. In this situation there is a plain vanilla terminology that $G$ is an "extension group". If you want to bring in mention of the kernel $\Lambda$ and the quotient $K$, then there is an unfortunately ambiguous terminology saying either that $G$ is an "extension of $K$ by $\Lambda$", or that $G$ is an "extension of $\Lambda$ by $K$". Another notation/terminology that is still evolving is to read the short exact sequence from left to right, leave the word "extension" out of it, and say that $G$ is a "$\Lambda$-by-$K$ group".

Once upon a time I was in the lucky situation of entitling a paper with complete lack of ambiguity as "A hyperbolic-by-hyperbolic hyperbolic group".


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