By Noel Brady
The origins of the be aware challenge are in workforce conception, decidability and complexity. yet during the imaginative and prescient of M. Gromov and the language of filling capabilities, the subject now affects the realm of large-scale geometry. This publication comprises money owed of many contemporary advancements in Geometric workforce concept and indicates the interplay among the observe challenge and geometry remains to be a vital topic. It comprises many figures, a number of routines and open questions.
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Extra info for The Geometry of the Word Problem for Finitely Generated Groups (Advanced Courses in Mathematics - CRM Barcelona)
The block f −1 ([− , 0]) contains a copy of the (− )-level set together with this copy of the circle coned oﬀ to v. This complex (the (− )-level set with coned oﬀ circle) is homotopy equivalent to f −1 ([− , 0]). An explicit homotopy equivalence is obtained by “pushing down” through the ambient 2-complex. 42 Chapter 2. Dehn Functions of Subgroups of CAT(0) Groups The next theorem formalizes these connections between the homotopy types of level sets through a vertex v on one hand, and the homotopy types of level sets just above (resp.
The following local condition guarantees when an Mκ -complex is locally CAT(κ). 11 (Link Condition). The Mκ -complex K is locally CAT(κ) if the link of every vertex in K is CAT(1). This is a recursive condition to check, because the links of vertices are piecewise spherical complexes of lower dimension. In order to see that these links are CAT(1), one needs to verify two things: 1. Check that the links are locally CAT(1), via a repeated application of the link condition to these piecewise spherical complexes.
The Geometry of the Word Problem for Finitely Generated Groups (Advanced Courses in Mathematics - CRM Barcelona) by Noel Brady