Errata for \Regular Neighbourhoods and Canonical Decompositions for Groups", Asterisque 289 (2003), by Peter Scott and Gadde A. Swarup

Invertible almost invariant sets The rst problem is in our treatment of almost invariant sets which are invertible (see De nition 2.12). Before discussing the details, we need to brie y recall the construction in chapter 3. We have a nitely generated group G with nitely generated subgroups H1; : : : ; Hn and, for 1 i n, we have a nontrivial Hi{almost invariant subset Xi of G. Recall that E denotes the set of all translates of all the Xi's and X i 's, and that an element U of E is isolated if it crosses no element of E. We construct the algebraic regular neighbourhood (X1; : : : ; Xn) in stages. First we consider the case when the Xi's are in good position. This means that if U and V are any elements of E and two of the four sets U ( ) \ V ( ) are small, then one must be empty. In Theorem 3.8, we describe a graph of groups structure (X1; : : : ; Xn) for G. If no Xi is simultaneously isolated and equivalent to an invertible almost invariant set, then (X1; : : : ; Xn) is de ned to be (X1; : : : ; Xn). If some Xi is isolated and is equivalent to an invertible almost invariant set Yi, we claimed (see the last ve lines of page 48) that we could replace each such Xi by an equivalent almost invariant set Zi such that Zi is not invertible and the new family is in good enough position. (Good enough position means that if U and V are any elements of E and two of the four sets U ( \V ( ) are small, then either one must be empty, or some element of E crosses both U and V .) This allowed us to de ne (X1; : : : ; Xn) in general. Unfortunately this claim is incorrect. The error is in the existence part of Lemma 3.14 and is already clear in the case when n = k = 1. Here is the statement of the existence part of Lemma 3.14 in this case.