We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev.As an application, we prove a group-theoretic analog of the GromovThurston 2π Theorem in the context of relatively hyperbolic groups.
We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling. Contents 1. The cusped space of a relatively hyperbolic group 3 2. Filling hyperbolic and relatively hyperbolic groups 5 3. Quasi-convexity 7 3.1. Induced peripheral structures 8 4. Proof of Theorem 0.6 11 4.1. Projections of geodesics to cusped spaces of quotients 11 4.2. The image of H is quasi-convex 14 4.3. Keeping g out of H 15 4.4. Height decreases under filling 16 4.5. The proof of Theorem 0.6 22 5. Conclusion 23 References 23
If G is a group, a pseudocharacter f: G-->R is a function which is "almost" a
homomorphism. If G admits a nontrivial pseudocharacter f, we define the space
of ends of G relative to f and show that if the space of ends is complicated
enough, then G contains a nonabelian free group. We also construct a
quasi-action by G on a tree whose space of ends contains the space of ends of G
relative to f. This construction gives rise to examples of "exotic"
quasi-actions on trees.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper26.abs.htm
Campus activists and others might refer to slights of one's ethnicity or other cultural characteristics as "microaggressions," and they m ight use various forums to publicize them . Here we exam ine this phenom enon by drawing from Donald Black's theories of conflict and from cross-cultural studies of conflict and morality. We argue th a t this behavior resembles other conflict tactics in w hich the aggrieved actively seek the sup port of third parties as well as those th a t focus on oppression. We identify the social conditions associated w ith each feature, and w e discuss how the rise of these condi tions has led to large-scale m oral change such as the em ergence of a victimhood culture th a t is distinct from th e honor cultures and dignity cultures of the past. Keywords microaggression -social control -morality -honor -dignity -victim hood * Bradley Campbell and Jason Manning contributed equally. They wish to thank Donald Black and Joseph Michalski for comments on an earlier version. COMPARATIVE SOCIOLOGY 13 (2014) 6 92-726
We give an alternate proof of Wise's malnormal special quotient theorem (MSQT), avoiding cubical small cancelation theory. We also show how to deduce Wise's Quasiconvex Hierarchy Theorem from the MSQT and theorems of Hsu and Wise and Haglund and Wise.
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