I'm reading a paper on group-C*-algebras and have difficulties finding a definition of the following:
Let P be a property of groups stable by taking subgroups. Let $T$ be a tree, and let $G\le\textrm{Aut}(T)$ be a subgroup such that fixators of edges in $G$ have P. Then for every $\xi\in\partial T$, the stabilizer of $\xi$ in $G$ is (locally P)-by-$\mathbf{Z}$ or locally P.
Later on there seems to be something similar going on:
Assume that P is a property of groups with the following properties: P is stable by taking subgroups, quotients and extensions; P = locally P; every group that is (locally finite)-by-P has P
The author continues to tell me that e.g. amenability, elementary amenability and local finiteness are such properties.
Could anyone explain to me the two expressions à la (locally X)-by-Y, please?
A group is P-by-Q if it has a normal subgroup with property P such that the quotient has Q.
(Assuming that P passes to subgroups): a group is locally P if all its finitely generated subgroups have P.
The meaning of (locally P)-by-Q follows.
[Note: what's above called "P-by-Q" is called "Q-by-P" by some authors, but in geometric group theory (as in the result above) the usual convention is the first one, and the statement about edge fixators you say, makes sense only with this convention.]