simple Lie group is hyperbolic if and only if the group is isogenous to (equivalently the associated symmetric space is the hyperbolic plane). An example of this is given by hyperbolic knot groups. Another is the Bianchi groups, for example . Properties Algebraic properties Hyperbolic groups satisfy the Tits alternative: they are either virtually solvable (this possibility is satisfied only by elementary hyperbolic groups) or they have a subgroup isomorphic to a nonabelian free group. Non-elementary hyperbolic groups are not simple in a very strong sense: if is non-elementary hyperbolic then there exists an infinite subgroup such that and are both infinite. It is not known whether there exists a hyperbolic group which is not residually finite. Geometric properties Non-elementary (infinite and not virtually cyclic) hyperbolic groups have always exponential growth rate (this is a consequence of the Tits alternative). Hyperbolic groups satisfy a linear isoperimetric inequality. Homological properties Hyperbolic groups are always finitely presented. In fact one can explicitly construct a complex (the Rips complex) which is contractible and on which the group acts geometrically so it is of type F∞. When the group is torsion-free the action is free, showing that the group has finite cohomological dimension. In 2002, I. Mineyev showed that hyperbolic groups are exactly those finitely generated groups for which the comparison map between the bounded cohomology and ordinary cohomology is surjective in all degrees, or equivalently, in degree 2. Algorithmic properties Hyperbolic groups have a solvable word problem. They are biautomatic and automatic. Indeed, they are strongly geodesically automatic, that is, there is an automatic structure on the group, where the language accepted by the word acceptor is the set of all geodesic words. It was shown in 2010 that hyperbolic groups have a decidable marked isomorphism problem. It is notable that this means that the isomorphism problem, orbit problems (in particular the conjugacy problem) and Whitehead's problem are all decidable. Cannon and Swenson have shown that hyperbolic groups with a 2-sphere at infinity have a natural subdivision rule. This is related to Cannon's conjecture. Generalisations Relatively hyperbolic groups Relatively hyperbolic groups are a class generalising hyperbolic groups. Very roughly is hyperbolic relative to a collection of subgroups if it admits a (not necessarily cocompact) properly discontinuous action on a proper hyperbolic space which is "nice" on the boundary of and such that the stabilisers in of points on the boundary are subgroups in