Semistability of Artin and Coxeter groups
In this paper all spaces are locally finite \(CW\)-complexes. If \(A\) is a subset of a space then \(St(A)\) is the union of all cells that intersect \(A\). Define \(St^n(A)\equiv St(St^{n-1}(A))\) for \(n\geq 1\) (here \(St^0(A)\equiv A\)). A continuous function \(f\) mapping the space \(X\) to the space \(Y\) is proper if for each compact set \(C\subset Y\), \(f^{-1}(C)\) is compact in \(X\). A ray in \(X\) is a proper map \(r\colon[0,\infty)\to X\). Two rays, \(r\) and \(s\), converge to the same end of \(X\) if for any compact subset \(C\subset X\) there is an \(N\geq 0\) such that \(r([N,\infty))\) and \(s([N,\infty))\) lie in the same component of \(X-C\). ``Converges to the same end'' is an equivalence relation on the set of all proper maps and the set of all equivalence classes is called the set of ends of \(X\). An end of \(X\) is semistable if any two rays of \(X\) converging to that end are properly homotopic. A space \(X\) is semistable at \(\infty\) if each end of \(X\) is semistable. A finitely presented group \(G\) is semistable at infinity if for some (equivalently, any) finite, connected complex \(Y\) with \(\pi_1(Y)=G\), the universal cover of \(Y\) is semistable at infinity. Semistability at infinity is a geometric invariant of finitely presented groups. If a finitely presented group \(G\) is semistable at infinity, then one can define \(\pi^\infty_1(G)\), the fundamental group at infinity for \(G\). It is conjectured that if \(G\) is a finitely presented group then \(H^2(G;\mathbb{Z} G)\) is free Abelian, but this is currently unknown, even for 2-dimensional duality groups. If \(G\) is semistable at infinity then \(H^2(G;\mathbb{Z} G)\) is free Abelian. In this paper, the author proves that all Artin and all Coxeter groups are semistable at infinity.
scientific article; zbMATH DE number 920030