Jul 22, 2019
USA 1984
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USA 1984
A finitely generated non-elementary hyperbolic group, i.e. a group whose cayley graph is a hyperbolic metric space, and which is infinite and has no infinite cyclic group of finite index, has a finite center.
The proof of this is in fact almost entirely geometric, and procceeds by showing that if the center has is infinite, then using the fact that it is unbounded in the cayley graph, we can produce an infinite cyclic group which is quasi-isometric to the entire group, and hence must be of finite index, so we get that a hyperbolic group with infinite center is virtually cyclic.
Math Fact
Noncommutative rings are more interesting than commutative rings
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