Ruth Charney

Bestvina's Normal Form Complex and the Homology of Garside Groups

AbstractA Garside group is a group admitting a finite lattice generating set

Contracting boundaries of CAT(0) spaces

\mathcal{D}

Metric characterizations of spherical and Euclidean buildings

. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π, 1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice

RECIPROCITY OF GROWTH FUNCTIONS OF COXETER GROUPS

\mathcal{D}

Automorphisms of two-dimensional RAAGS and partially symmetric automorphisms of free groups

, and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π, 1)s enjoy Bestvinas weak nonpositive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually Abelian.

Convexity of parabolic subgroups in Artin groups

In 1977, Vogtmann [lo] proved that the groups O,.(k) are homology stable for any field k, k# [F,. That is, the natural homomorphisms H;(O,.(k); Z)-+ H;(O n + I,n+ i(k); Z) are isomorphisms for n sufficiently large. Recently, Betley [4] has extended this theorem to include local rings k. It is generally expected (and occasionally assumed) that the same is true if k=Z and, most likely, for a much larger class of rings as well, but no proof of this exists in the literature. As this fact, for k= Z’, is an essential ingredient in a forthcoming joint paper with R. Lee, I have undertaken to present a proof. The main result contained here is the homology stability of O,.(A) and SpZn(A), with twisted as well as untwisted coefficients, in the case where A is a Dedekind domain. Much of the discussion, however, is carried out in a broader context in the hopes that it can be used for still further generalizations. In particular, it is conjectured that stability holds for any finite algebra A over a commutative ring k with noetherian maximal spectrum, and so as much of the theory as possible is developed in this setting. The bulk of the work, as always in proving homology stability, is the construction of highly connected simplicial complexes on which the groups act with ‘nice’ stabilizer subgroups. In this case, we use the simplicial complex arising from the partially ordered set of ‘hyperbolic unimodular’ sequences (see Section 3) which has the nicest possible stabilizer subgroups, namely lower-dimensional orthogonal or symplectic groups. The proof of the connectedness of this complex occupies Sections l-3. The methods used owe much to the work Maazen [8] and Van der Kallen [7]. The last section, Section 4, contains the stability theorems.