1.2. A SCHEME FOR UNDERSTANDING GROUPS 5 1.2. A Scheme for Understanding Groups The above discussions are instances of partial success in implementing the following “grand plan” for understanding many groups: ● Find codimension-1 subgroups in a group G. ● Produce the dual CAT(0) cube complex ̃ upon which G acts. ● Verify that G acts properly and relatively cocompactly on ̃ by examining the extrinsic nature of the codimension-1 subgroups. ● Consequently G is the fundamental group of a nonpositively curved cube complex C = G/ ̃. (Or C is an orbihedron if G has torsion.) ● Find a finite covering space ̂ of C, such that ̂ is special. ● The specialness reveals many structural secrets of G. For instance, G is linear since it embeds in SLn(Z), and the geometrically well- behaved subgroups of G are separable. There is much work to be done to determine exactly when the above plan can be applied, and there are certainly groups where the plan is im- possible to implement – e.g. any nonlinear group. However, when the plan is successful, it provides a very useful viewpoint. We conclude that in many cases, especially when G has comparatively few relators, we see that: Though G might arise as the fundamental group of a small 2-complex or 3-manifold, in many cases one should sacrifice this small initial presentation in favor of a much larger and higher-dimensional object that is a nonposi- tively curved special cube complex, and has the advantage of being far more organized, thus revealing important structural aspects of G.
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