Monthly Archives: February 2011

The Rips Construction I

Given a finitely presented group , Rips constructed a hyperbolic group and a surjection with finitely generated kernel . In this post and the next we will see how the Rips construction has been used to produce draconic subgroups of … Continue reading

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Well behaved subgroups of non-positively curved groups

Our main source for this post is Bridson and Haefliger’s book [BrH]. This post concerns contexts in which the subgroups of non-positively curved groups are well behaved. We have already considered hyperbolic groups; now we turn to the various classes … Continue reading

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Quasi-convex subgroups of hyperbolic groups

Our main source for this post is Bridson and Haefliger’s book [BrH]. However, the material has origins in Alonso, T. Brady, Cooper, Ferlini, Lustig, Mihalik, Shapiro and Short (also ed.), Gersten and Short, Ghys and de al Harpe, Sur les … Continue reading

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Non-positively curved groups IV – quadratic isoperimetric functions

The main sources for this post are the slides for Riley’s talk “How wild can a group with quadratic Dehn function be?” and Section 4.1 of Sapir’s article “Asymptotic invariants, complexity of groups and related problems”. In Euclidean space, a … Continue reading

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Non-positively curved groups III – combable, automatic, semi-hyperbolic, and all that

This post is based, in part, on portions of Martin Bridson’s 2006 ICM article. As we’ve described, the metric in a CAT(0) space is convex. We now turn to what this means for a group which acts properly cocompactly by … Continue reading

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Non-positively curved groups II – hyperbolic groups

In this post and the next we will survey an assortment of intrinsic properties groups can enjoy that bear vestiges of non-positive curvature. We will focus here on hyperbolicity, which stands out as a compelling notion of negative curvature for … Continue reading

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Non-positively curved groups I – CAT(0) and CAT(-1) groups

This post is partially based on portions of Martin Bridson’s 2006 ICM article. In the last post we explained what it means to say a space is CAT(0) or CAT(-1). Now we turn to groups. CAT(0) and CAT(-1) groups A … Continue reading

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