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.
In a 2010 paper, it was shown 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.
Read more about this topic: Hyperbolic Group
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)