A refinement of the simple connectivity at infinity for groups

Abstract

We give another proof for a result of Brick ([2]) stating that the simple connectivity at infinity is a geometric property of finitely presented groups. This allows us to define the rate of vanishing of π ∞ 1 for those groups which are simply connected at infinity. Further we show that this rate is linear for cocompact lattices in nilpotent and semi-simple Lie groups, and in particular for fundamental groups of geometric 3-manifolds.

On groups with linear sci growth

“…In contrast with the abundance of equivalence classes of geometric invariants of finitely presented groups, (like group growth, Dehn functions or isodiametric functions) the metric refinements of topological properties seem highly constrained. We already found in [10] that many cocompact lattices in Lie groups and in particular geometric 3-manifold groups have linear sci. The aim of this paper is to further explore this phenomenon by considerably enlarging the class of groups with linear sci.…”

“…It is proved in [10] that the (rough) equivalence class of V X (r) is a quasi-isometry invariant. In particular, if G is sci, then the (rough) equivalence class of the real function V G = V XG is a quasiisometry invariant of the group G, where X G is the universal covering space of any finite complex X G , with π 1 (X G ) = G. If G is sci and V G is a linear function we will say that G has linear sci.…”

“…There is a well-defined notion of topological fundamental group at infinity associated to a semistable end of a group (see [11]). Now, following [10] we consider the following metric refinement of the semistability: Definition 7. Let X be a non-compact metric space, e an end of X and γ a ray converging to e. The semistability growth function S e (r) is the infimal N with the following property: for any R ≥ N and any loop l based on γ which lies in X − B(N ) there exists a homotopy rel γ supported in X − B(n) which moves l to a loop in X − B(R).…”

“…Although this depends on the chosen generating set the different word metrics are quasi-isometric. In [10] we enhanced the topological sci notion in the case of groups by taking advantage of this metric structure. [10]) is the infimal N (r) with the property that any loop in the complement of the metric ball B(N (r)) of radius N (r) (centered at the identity) bounds a 2-disk outside B(r).…”

On the wgsc and qsf tameness conditions for finitely presented groups

“…One can show easily that the rough equivalence class of f G;P .r/ depends only on the group G and not on the particular presentation, following [8] and [24]. We will write it as f G .r/.…”