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- Thompson_groups abstract "This page is about the infinite Thompson groups F, T and V. For the sporadic finite simple group Th see Thompson sporadic group.In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted , which were introduced by Richard Thompson in some unpublished handwritten notes in 1965. Of the three, F is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group.The Thompson groups, and F in particular, have a collection of unusual properties which have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group F is not simple but its derived subgroup [F,F] is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2.The conjecture that F is not amenable was made by R. Geoghegan in 1979 --- see Cannon-Floyd-Parry article cited in the references below. (R. Thompson has stated in a lecture that he considered this question in the late 1960's.) Its current status is controversial: E. Shavgulidze published a paper in 2009 in which he claimed to prove that F is amenable, but an error was found, as is explained in the MR review. On the other hand Azer Akhmedov and Leva Beklaryan independently posted preprints claiming that F is not amenable, though as of 2013 these proofs have not been accepted as correct. The status of the Conjecture is still considered to be open.It is known that F is not elementary amenable. If F is not amenable, then it would be another counterexample to the long-standing but recently disprovedvon Neumann conjecture for finitely-presented groups, which suggested that a finitely-presented group is amenable if and only if it does not contain a copy of the free group of rank 2.Higman (1974) introduced an infinite family of finitely presented simple groups, including Thompson's group V as a special case.".
- Thompson_groups thumbnail AA_Tree_Skew2.svg?width=300.
- Thompson_groups wikiPageExternalLink books?id=LPvuAAAAMAAJ.
- Thompson_groups wikiPageExternalLink rtx110801112p.pdf.
- Thompson_groups wikiPageExternalLink cfp.pdf.
- Thompson_groups wikiPageID "612245".
- Thompson_groups wikiPageRevisionID "600984858".
- Thompson_groups hasPhotoCollection Thompson_groups.
- Thompson_groups subject Category:Infinite_group_theory.
- Thompson_groups comment "This page is about the infinite Thompson groups F, T and V. For the sporadic finite simple group Th see Thompson sporadic group.In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted , which were introduced by Richard Thompson in some unpublished handwritten notes in 1965.".
- Thompson_groups label "Thompson groups".
- Thompson_groups label "Группы Томпсона".
- Thompson_groups label "湯普森群".
- Thompson_groups sameAs m.02wh0s.
- Thompson_groups sameAs Q4150823.
- Thompson_groups sameAs Q4150823.
- Thompson_groups wasDerivedFrom Thompson_groups?oldid=600984858.
- Thompson_groups depiction AA_Tree_Skew2.svg.
- Thompson_groups isPrimaryTopicOf Thompson_groups.
