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Semihyperbolic groups. (English) Zbl 0823.20035

We define semihyperbolicity, a condition which describes non-positive curvature in the large for an arbitrary metric space. This property is invariant under quasi-isometry. A finitely generated group is said to be weakly semihyperbolic if when endowed with the word metric associated to some finite generating set it is a semihyperbolic metric space. Such a group is of type \(FP_{\infty}\) and satisfies a quadratic isoperimetric inequality. We define a group to be semihyperbolic if it satisfies a stronger (equivariant) condition. We prove that this class of groups has strong closure properties. Word-hyperbolic groups and biautomatic groups are semihyperbolic. So too is any group which acts properly and cocompactly by isometries on a space of nonpositive curvature. A discrete group of isometries of a 3-dimensional geometry is not semihyperbolic if and only if the geometry is Nil or Sol and the quotient orbifold is compact. We give necessary and sufficient conditions for a split extension of an abelian group to be semihyperbolic; we give sufficient conditions for more general extensions. Semihyperbolic groups have a solvable conjugacy problem. We prove an algebraic version of the flat torus theorem; this includes a proof that a polycyclic group is a subgroup of a semihyperbolic group if and only if it is virtually abelian. We answer a question of Gersten and Short concerning rational structures on \(\mathbb{Z}^ n\).

MSC:

20F65 Geometric group theory
57M07 Topological methods in group theory
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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