Download e-book for kindle: Geometric Group Theory: Volume 1 by Graham A. Niblo, Martin A. Roller

By Graham A. Niblo, Martin A. Roller

ISBN-10: 051166186X

ISBN-13: 9780511661860

ISBN-10: 0521435293

ISBN-13: 9780521435291

Those volumes include survey papers given on the 1991 foreign symposium on geometric staff concept, they usually symbolize a few of the most modern considering during this sector. some of the world's top figures during this box attended the convention, and their contributions conceal a large variety of subject matters. quantity I includes experiences of such topics as isoperimetric and isodiametric capabilities, geometric invariants of a teams, Brick's quasi-simple filtrations for teams and 3-manifolds, string rewriting, and algebraic evidence of the torus theorem, the type of teams appearing freely on R-trees, and lots more and plenty extra. quantity II is composed completely of a flooring breaking paper through M. Gromov on finitely generated teams.

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Example text

1) Fir,') := {c E F I v(c) > r}, r E R. A G-module A is said to be of type FPm if it admits a free resolution F 0, with finitely generated rn-skeleton F(m) := $m Fi. In order to mimic the i=o topological situation we may assume that F is given with a ZLG-basis X C F Geometric invariants of a group with the property that 0 31 U. It is clear that such resolutions always exist and we call them admissible. As naive valuations can achieve arbitrary prescribed values on X we can pick naive valuations vi : F; -> R00 in such a way that v;_18 > vi, for all i.

The homotopical geometric invariant E*(G) Let G be a group. By an Eilenberg-MacLane complex for G, or a K(G, 1), we mean a connected CW-complex with fundamental group G and vanishing higher homotopy groups. Following C. T. C. Wall we say that G is of type F,,,, m E No, if G admits an Eilenberg-MacLane complex Y with finite m-skeleton Ym. Throughout the paper we will assume that the group G is of type Fm for some fixed m, and that X is the universal covering complex of a K(G, 1) with finite m-skeleton.

It is conceivable that, for metabelian groups G, the invariant E'(G) contains the information whether G is of type Fm (resp. FPm). We have reasons to think that the following might be the answer. Conjecture. [B 81] A metabelian group G is of type Fm if and only if the complement of E' (G) in Hom(G,1R) has the property that every subset of m non-zero points is contained in an open half space of Hom(G, IR). Note that Theorem F establishes the conjecture for m = 2. The conjecture has been proved to hold true for metabelian groups of finite rank (Aberg[A]).

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