Relative ends and duality groups

Analogues of Cayley graphs for topological groups

“…The notion of ends of pairs of groups appeared first in papers by Houghton [23] and Scott [48]. A variant of the idea of ends relative to a subgroup was introduced by Kropholler and Roller in [31]. Here we use the name coends for Krophollers and Rollers concept and use the following reformulation, due to Bowditch [4], as a definition.…”

Relative ends, ℓ2-invariants and property (T)

“…See [13] for examples. The end invariantẽ(G, H) mentioned above is a generalisation of Scott's end invariant and was introduced by Kropholler and Roller, [7], in their study of the algebraic torus theorem for Poincaré duality groups. We will state the definition ofẽ(G, H) in Section 1, but note here that in particular if e(G, H) 2 thenẽ(G, H) 2 as required.…”

On the coarse-geometric detection of subgroups

“…In factẽ(G, H ) is closely related to e(G, H ). For instance, in [13], Kropholler Recall that a subset of G is said to be H -finite if it is contained in finitely many cosets Hg, and that this condition is equivalent to the subset being contained in some uniform neighborhood of H . We say that a subset is H -infinite if it is not H -finite.…”

“…13 Let G and H be as in Theorem 4.3. Then for any subgroup H 1 of H and any infinite index subgroup K of H 1 , e(G, K ) = 1.…”