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Semihyperbolic Groups
Author(s) -
Alonso Juan M.,
Bridson Martin R.
Publication year - 1995
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-70.1.56
Subject(s) - mathematics , abelian group , isoperimetric inequality , pure mathematics , group (periodic table) , finitely generated group , equivariant map , conjugacy class , quotient , discrete group , isometry group , hyperbolic group , invariant (physics) , combinatorics , finitely generated abelian group , mathematical analysis , hyperbolic manifold , chemistry , organic chemistry , mathematical physics , hyperbolic function
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 x 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 non‐positive 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 Z n .