1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.3. Hyperbolic groups 185 distance) to an honest geodesic. To make this precise, we must define "close to being geodesic." For ...

186 5. Exact Groups and Related Topics and o; is a (C, r)-quasigeodesic path, the length of a' is at most 6CDo + r. By joining / ...

5.3. Hyperbolic groups 187 Lemma 5.3.8. There exists a constant C = C(K) > 0 with the following property: For any two equival ...

188 5. Exact Groups and Related Topics particular, countable. We leave it as an exercise to check that every x EK can be connect ...

5.3. Hyperbolic groups 189 Now, it is easy to check that {U(z, R)}R>O defines a (not necessarily open) neighborhood basis and ...

190 5. Exact Groups and Related Topics Note that for every x, y EK and k, l, we have {z E aK: y E 8(x,z,l,k)} = LJ{T(x', y) : x' ...

5.3. Hyperbolic groups 191 Since (D(2n + d))-^2 d/n----+ 1 as n----+ oo, this proves the claim. Now we fix a base point o E K, s ...

192 5. Exact Groups and Related Topics We've seen that the left-translation action of a hyperbolic group r on .e= (r) is amenabl ...

5.4. Subgroups of Lie groups 193 5.4. Subgroups of Lie groups Discrete subgroups of Lie groups provide lots of examples of amena ...

194 5. Exact Groups and Related Topics define a Borel map jj,: X----+ Prob(G) by P,x = a-(x)v. One checks that max sup llsP,x - ...

Coarse metric spaces 195 of finite symmetric subsets of r such that LJ En = r and EnEm c En+m for every n, m. It is not ha ...

196 5. Exact Groups and Related Topics (1) X has property A; (2) for any R > 0 and E > 0, there exists a positive definite ...

5.5. Coarse metric spaces 197 finite direct sum of the nuclear C* -algebras of the form £^00 ® Mk ( <C). For each x EX, we de ...

198 5. Exact Groups and Related Topics for x, z EX. Then, for some S > 0, we have supp (x c Bs(x) for all x EX. Moreover, ll( ...

5.5. Coarse metric spaces 199 In particular, we have llf(x)ll < oo. On the other hand, if d(x,y) > maxk~n sk, then n llf(x ...

200 5. Exact Groups and Related Topics Remark 5 .. 5,10. Gromov's infamous construction of nonexact groups yields examples whose ...

5.6. Groupoids 201 (although there are important examples of non-Hausdorff locally compact groupoids). By the Hausdorff property ...

202 5. Exact Groups and Related Topics a pseudogroup. If ,...., is an equivalence relation on X, then the collection of all part ...

5.6. Groupoids 203 The topology of Gu is the relative product topology, where I is discrete. The groupoid Gu is an etale locally ...

204 5. Exact Groups and Related Topics fore, 'fJ E Cc(G) and f E Co(G(o)). Observe that (e, 'fJ) E Cc(G(o)) is the restriction o ...