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Large scale geometry of commutator subgroups. (English) Zbl 1195.20046
Summary: Let $$G$$ be a finitely presented group, and $$G'$$ its commutator subgroup. Let $$C$$ be the Cayley graph of $$G'$$ with all commutators in $$G$$ as generators. Then $$C$$ is large scale simply connected. Furthermore, if $$G$$ is a torsion-free nonelementary word-hyperbolic group, $$C$$ is one-ended. Hence (in this case), the asymptotic dimension of $$C$$ is at least 2.
##### MSC:
 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups 57M07 Topological methods in group theory 20F05 Generators, relations, and presentations of groups 20F12 Commutator calculus 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 55M10 Dimension theory in algebraic topology
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##### References:
 [1] M Bestvina, K Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002) 69 · Zbl 1021.57001 [2] R Brooks, Some remarks on bounded cohomology, Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 53 · Zbl 0457.55002 [3] D Calegari, scl, to appear in MSJ Memoirs, Math. Soc. of Japan · Zbl 1187.20035 [4] D Calegari, K Fujiwara, Stable commutator length in word-hyperbolic groups · Zbl 1227.20040 [5] M Dehn, The group of mapping classes, Springer (1987) 135 · Zbl 1264.01046 [6] D B A Epstein, K Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997) 1275 · Zbl 0884.55005 [7] K Fujiwara, K Whyte, A note on spaces of asymptotic dimension one, Algebr. Geom. Topol. 7 (2007) 1063 · Zbl 1141.51010 [8] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) 5 · Zbl 0516.53046 [9] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015 [10] M Gromov, Asymptotic invariants of infinite groups, London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1 · Zbl 0841.20039 [11] J R J Groves, Rewriting systems and homology of groups, Lecture Notes in Math. 1456, Springer (1990) 114 · Zbl 0732.20032 [12] R C Kirby, The topology of $$4$$-manifolds, Lecture Notes in Math. 1374, Springer (1989) · Zbl 0668.57001 [13] J F Manning, Geometry of pseudocharacters, Geom. Topol. 9 (2005) 1147 · Zbl 1083.20038 [14] J Stallings, Group theory and three-dimensional manifolds, Yale Math. Monogr. 4, Yale University Press (1971) · Zbl 0241.57001
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