The boundary of negatively curved groups

Abstract

Every finitely generated group r can be endowed with the word metric d (invariant under left translations) with respect to a given finite generating set.Following Gromov [Gr] we say that r is negatively curved provided there exists a (large) number t5 such thatfor all x, y, Z E r. Here (x· y) stands for the "overlap function"Gromov's article [Gr] When r has torsion, we can construct a highly connected polyhedron with free and cocompact r-action. For integers d and m, let P(d, m, f) be the subcomplex of the m-fold join r * ... * r consisting of those simplices whose vertices are within d in r. The natural map P(d, m, f) ---+ Pd(f) has the property that the preimage of each simplex is (m -2)-connected. It follows that for large d, P(d, m, f) is an (m -2)-connected free r-complex with compact quotient. This shows for example that r is a group of type F P 00 • Furthermore, for any ring R (in practice R = Z or Q) the compactly supported cohomology H: (P(d, m, r); R), which is (by definition, see [Br]) isomorphic

Continuous control and the algebraic L-theory assembly map

“…More generally, virtually polycyclic groups satisfy these conditions, since we can also take R n for some n to be our universal space [Wil00]. Gromov's word hyperbolic groups also satisfy the conditions by taking a certain compactification of the Rips complex [BM91,MS02].…”

Detecting free splittings in relatively hyperbolic groups

“…condition of Bestvina and Mess [1], which can be used, if the boundary has no cut points, to discover whether it is connected (Proposition 5.1). Finally, in Section 5 we present the algorithms which prove Theorems 1.2, 1.3 and 1.4.…”

“…The first two types are called horizontal, and the last type is called vertical. • H (2) contains three kinds of 2-cells: (1) If γ ⊂ H (1) is a circuit composed of three horizontal edges, then there is a 2-cell (a horizontal triangle) attached along γ.…”

“…Up to changing some elements e of the basis A 1 into −e, we can assume that all coordinates of a 1 Proof. Note that in the proof below we appeal to Lemma 2.15.…”

“…Since the distance in X between x and y is at most 2M + 3, we have d i (x, y ) ≤ 2 2M +3 . The intersection of level i with σ 1 consists of two points, x and another point p 1 . The intersection of σ 2 with level i consists of points y and p 2 .…”