# Who was Richard Thompson?

There is family of famous groups with unusual group-theoretic properties due to a mathematician called Richard Thompson that are widely studied in group theory. The papers on these groups and the Wikipedia entry simply say that these were written down in some unpublished hand-written notes of Thompson in the 1960's. A search on Google / Wikipedia / MathSciNet has not revealed anything about who Richard Thompson was, except a possible implication that Thompson was motivated by questions coming from logic. Does anyone have any information?

• @njuffa : thanks for all the info. This is very interesting. I really appreciate the effort. Jul 9 at 6:22
• Not the same as J.G.Thompson, of the Feit-Thompson theorem. Jul 10 at 0:23
• R. Thompson is a former graduate student of Ralph McKenzie in UC Berkeley Jul 10 at 6:58
• Seeing that there are still actively working mathematicians that have collaborated with him, cross posting on mathoverflow might be worthwhile, hoping for the non trivial chance of an answer from firsthand experience. Jul 11 at 7:29
• When I was a graduate student at Berkeley 2010-2016, Richard Thompson was a regular attendee of the Berkeley Logic Colloquium. I assume he still attends (when the pandemic situation allows). Jul 12 at 20:28

I was only able to find fragmentary information on Thompson. A good starting point is his PhD thesis, which gives us his full name:

• Richard Joseph Thompson, Transformational Structure of Algebraic Logics. PhD thesis, University of California, Berkeley, 1979

A partial scan of the thesis is publicly available free of cost. From this we learn that Thompson received an A.B. degree from San Jose State in 1965, that he was accepted into the PhD program at Berkeley in 1973, and that his degree was conferred on June 16, 1979. His thesis advisor was Ralph McKenzie. This agrees with the entry for McKenzie at the Mathematics Genealogy Project. Curiously, there is no mention of R. J. Thompson in the La Torre college yearbooks of San Jose State from the years 1964 and 1965.

Thompson's handwritten notes are inconsistently referenced in the literature, with some dating them to 1965 (e.g. David Matthew Robertson, Conjugacy and centralisers in Thompson’s group $$T$$, PhD thesis, Newcastle University, 2018), while others date them to 1973 (e.g. John Donnelly, "Ruinous subsets of Richard Thompson’s group $$F$$," Journal of Pure and Applied Algebra 208 (2007) 733–737). In many instances the notes are referred to as "widely circulated". What are claimed to be scans of Thompson's original hand-written notes can be found here: part 1, part 2.

The first proper publication of the material appears to be in R. McKenzie and R. J. Thompson, "An elementary construction of unsolvable word problems in group theory", in Word Problems, Proc. Conf. Irvine 1969 (edited by W. W. Boone, F. B. Cannonito, and R. C. Lyndon), Studies in Logic and the Foundations of Mathematics, vol. 71, North-Holland, Amsterdam, 1973, pp. 457–478.

There are only a few other papers by R. J. Thompson, some of which appeared in somewhat obscure Eastern European publications. Thompson did a brief stint at the Institute for Advanced Studies in Princeton in 1974-1975. Membership lists of the Association for Symbolic Logic published in The Journal of Symbolic Logic, Vol. 45, No. 4, Dec. 1980 and The Journal of Symbolic Logic, Vol. 47, No. 4, Dec. 1982 show him as a member of that organization. Publications in 1988 and 1993 indicate that Thompson was affiliated with the Mathematical Institute of the Hungarian Academy of Sciences, Budapest:

• Richard J. Thompson, "Noncommutative cylindric algebras and relativizations of cylindrical algebras", Bulletin of the Section of Logic 17 (2):75-81 (1988) (scan online)

• H. Andréka and R. J. Thompson, "A Stone type representation theorem for algebras of relations of higher rank", Transactions of the American Mathematical Society, Vol. 309, No. 2, October 1988, pp. 671-682 (scan online)

• Richard J. Thompson, "Complete description of substitutions in cylindric algebras and other algebraic methods." Algebraic Methods in Logic and in Computer Science, Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Vol. 28, 1993, pp. 327-342. (scan online)

The last professional trace of R. J. Thompson coincides with the last known year of publication, when he was listed as a participant, but not presenter, at a workshop/conference: Universal Algebra and Category Theory Conference, Berkeley, July 12-23, 1993.

• Wait, he was admitted to the Berkley PhD program in 1973, then did a stint at the IAS in Princeton in 1974-1975, and then went back to Berkley to finish his PhD in 1979? Doesn't this all seem a bit odd and out of order? Jul 10 at 13:33
• @RBarryYoung: Most likely, McKenzie was spending a sabbatical year at IAS and took his PhD student (Thompson) with him to Princeton, to continue advising. This is completely normal. Jul 10 at 14:17

Thompson contributed to group theory, and algebraic logic. The original motivation for his calculations/constructs was algebraic logic. His thesis and all but two of his papers were on sets of laws that characterized certain types of algebraic logics with an eye to finding such sets that are finite.

The first setting was a system of expression manipulations (duplication or elimination of a symbol, shifting parentheses, permuting symbols) which gave a monoid whose group of units was the group now known as V. He proved the group infinite and finitely presented and when advised to find normal subgroups was able to show there are none. This is the subject of his handwritten notes and is his most well known result in group theory. In those notes Thompson establishes the basics of the three groups now known as F, T, and V. The first part of the expository paper of Cannon, Floyd and Parry on the Thompson groups is a detailed expansion of those notes.

His two papers not in algebraic logic are the one mentioned by njuffa and one paper by himself extending an embedding theorem of Boone and Higman.

Embeddings into finitely generated simple groups which preserve the word problem, in Word Problems, II, (S. I. Adian, W. W. Boone, G. Higman, eds.), North-Holland (1980) 401-441.

That last paper is also quite well known. Both papers in group theory contain pieces of the basic information about the groups F and V.

His work in algebraic logic is highly regarded by the algebraic logic community. For most of his life Thompson worked through an agency for K-12 students with reading and math difficulties in the bay area around San Jose, CA.

By the way, the scanned notes in njuffa's response contains a common, often propogated error. The third page of Part 1 (it starts with $$C_n^m$$) should actually be the fifth page. No other rearrangement is necessary.

Richard Thompson is alive and retired in California.

• Has anyone made a LaTeX version of the handwritten notes? Seems quite an easy thing to do... Jul 11 at 1:36
• @Carl-FredrikNybergBrodda: The Thompson notes are complete in parts and are very sketchy in others. The (1996) Cannon-Floyd-Parry article in Enseignement fills in all the gaps, gives a good history in the first couple of pages, and adds material, with attributions where needed. Of course if it is deemed of historical interest ... Jul 11 at 15:31
• Thanks @MattBrin for this info. I appreciate the texture that all of this provides. Maybe I should have added that my initial reason for being interested in Richard Thompson is that I was watching a 2018 ICM lecture of Andres Navas, who referred to the groups of Richard Thompson. I was already aware of the group theorist John Thompson, and was interested in trying to understand if this could have been the same person. When I tried to use basic search methods to find out about R. Thompson, I didn't come up with much at all. Jul 11 at 17:32