Bieri-strebel Valuations (of Finite Rank)
نویسنده
چکیده
Page Introduction 269 Notation and terminology 271 I. Basic facts 271 1. Review of IM 271 2. Review of AM 273 3. Solvable groups, aG, finite Priifer rank, FP 2 , and HNN-extensions . 276 4. Matrix computation of AM in the finite rank case . . . . 277 5. Finiteness conditions: F P m and finite presentation . . . . 278 6. The properties m-tameness and w-domestication 280 II. Proof that Hm{G, Y\ RG) zero implies m-domestication . . . . 2 8 1 Outline of proofs 281 1. The submodule 282 2. Invertible fractional ideals inside the submodule 283 3. The 'fac' function 284 4. Rudiments of an estimate 287 5. A map through HJG, f ] RG) and a relative 'fac' . . . . 2 8 8 6. The estimate for invertible fractional ideals 288 7. The estimate 290 8. Upper bound for v 291 9. Main theorem 292 10. Final statement 293 III. Proof that w-tame implies F P m 293 Outline and start of proofs 293 1. Reduction to trivial extensions 294 2. Construction of the space Y, and the group ^ 295 3. Computat ion of the w-connectedness of Y 299 IV. Conclusion 301 1. Metabelian groups 301 2. Solvable groups 302 References 303
منابع مشابه
Finitely Presented Extensions by Free Groups
We prove here that a finitely presented group with a free quotient of rank n is an HNN-extension with n stable letters of a finitely generated group where the associated subgroups are finitely generated. This theorem has a number of consequences. In particular, in the event that the free quotient is cyclic it reduces to an elementary and quick proof of a theorem of Bieri and Strebel. 1. Finitel...
متن کاملBieri–neumann–strebel–renz Invariants and Homology Jumping Loci
We investigate the relationship between the geometric Bieri–Neumann– Strebel–Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds ca...
متن کاملClosed 1-forms in Topology and Geometric Group Theory
In this article we describe relations of the topology of closed 1-forms to the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CWcomplexes and show that many properties of the group theoretic version have analogous statements. In particular we show the relation between Sigma invariants and finiteness properties of cert...
متن کاملInfinite Presentability of Groups and Condensation
We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that ...
متن کاملThe Sigma Invariants of the Lamplighter Groups
We compute the Bieri-Neumann-Strebel-Renz geometric invariants, Σn, of the lamplighter groups Lm by using the Diestel-Leader graph DL(m,m) to represent the Cayley graph of Lm.
متن کامل