Logic seminar, Fall 2021

The Cornell Logic Seminar (MATH 7810) is both a reading seminar for students (Tuesdays) and one which features research talks by outside speakers, faculty, and students (Fridays). The logic seminar page contains more complete information for past and upcoming seminar talks (including abstracts).

Logic and Thompson's groups

The reading seminar in Fall 2021 will a variety of papers on Richard Thompson's groups \(F\), \(T\), and \(V\) as well as their relatives. This includes the Brin-Thompson groups \(nV\) which serve as higher dimensional generalizations of \(V\) and well as groups of piecewise linear and piecewise projective homeomorphisms of the interval and circle — for instance Stein's groups \(F_{p_1,\ldots,p_k}\) and Cleary's golden ratio \(F\). No prior knowledge of either Thomson's groups or logic will be required.

The semester will begin with an introduction to Thompson's groups taken from Cannon, Floyd, and Parry's Introductory Notes on Richard Thompson's Groups. The topics will all be tangential to logic or set theory in some way. They will be chosen from:

• Thompson's original use of his groups to exhibit finitely presented groups with unsolvable word problem.
• The undecidability of the torsion and conjugacy problems in the Brin-Thompson group \(2V\).
• The coNP-completeness of the word problem for \(2V\) as well as other complexity theory aspects of word problems of the Brin-Thompson group.
• The conjectured relationship between \(V\) and those groups with a co-context free word problem (\(co\mathcal{CF}\)).
• Zimmer's theory of amenable equivalence relations and the role that it plays in establishing the nonamenability of certain groups of piecewise projective homeomorphisms of the interval.
• The role that transducers and automata play in studying Thompson-like groups.
• The Brin-Sapir Conjecture and the hierarchy of finitely generated subgroups of \(F\). This includes the interplay between ordinal arithmetic and the elementary finitely generated subgroups of \(F\), as well as its metamathematical consequences.
• Obstructions to embedding subgroups of \(\mathrm{PL}_o I\) and \(\mathrm{PP}_o I\) into \(F\).
• Uses of set theory and logic in studying the amenability problem for \(F\).
• Reconstruction theorems for groups of homeomorphims of locally compact spaces.

A list of references for the semester is given below. As there is too much material to cover in the course of the semester, the exact list of topics covered will be determined based on the interests of the participants.

While the seminar will primarily be held in person, some talks will be given via Zoom (typically in the case of outside speakers). These will be recorded and posted under the Zoom menu item (when viewing the recordings, it may be necessary to login to Cornell Zoom via SSO).

Course expectations

Students may enroll in this course for credit under the SX/UX option. Students who are enrolled in the course are expected to share the task of presenting the topic material. While this will depend on the number of students enrolled, this typically involves presenting 2–3 lectures from a single paper. Students enrolled in the seminar are expected to regularly attend talks related to the topic (including those topic-related outside talks).

References

Note: some files linked below may differ slightly from the published version which is cited.

• Altinel, Tuna and Muranov, Alexey Interprétation de l'arithmétique dans certains groupes de permutations affines par morceaux d'un intervalle.. J. Inst. Math. Jussieu 8 (2009), no. 4, 623–652. (link to is to English translation arXiv:0807.1079).
• Belk, James; Bleak, Collin Some undecidability results for asynchronous transducers and the Brin-Thompson group \(2V\). Trans. Amer. Math. Soc. 369 (2017), no. 5, 3157–3172.
• Birget, Jean-Camille The groups of Richard Thompson and complexity. International Conference on Semigroups and Groups in honor of the 65th birthday of Prof. John Rhodes. Internat. J. Algebra Comput. 14 (2004), no. 5–6, 569–626.
• Birget, Jean-Camille The word problem of the Brin-Thompson group is 𝖼𝗈𝖭𝖯-complete. J. Algebra 553 (2020), 268–318.
• Bleak, Collin; Brin, Matthew G.; Moore, Justin Tatch Complexity among the finitely generated subgroups of Thompson's group. J. Comb. Algebra 5 (2021), no. 1, 1–58.
• Bleak, Collin; Brin, Matthew; Kassabov, Martin; Moore, Justin Tatch; Zaremsky, Matthew C. B. Groups of fast homeomorphisms of the interval and the ping-pong argument. J. Comb. Algebra 3 (2019), no. 1, 1–40.
• Bleak, Collin; Matucci, Francesco; Neunhöffer, Max Embeddings into Thompson's group \(V\) and \(co\mathcal{CF}\) groups. J. Lond. Math. Soc. (2) 94 (2016), no. 2, 583–597.
• Brin, Matthew G. Higher dimensional Thompson groups. Geom. Dedicata 108 (2004), 163–192.
• Brin, Matthew G. The ubiquity of Thompson's group \(F\) in groups of piecewise linear homeomorphisms of the unit interval. J. London Math. Soc. (2) 60 (1999), no. 2, 449–460.
• Brin, Matthew G.; Squier, Craig C. Groups of piecewise linear homeomorphisms of the real line. Invent. Math. 79 (1985), no. 3, 485–498.
• Burillo, José; Matucci, Francesco; Ventura, Enric The conjugacy problem in extensions of Thompson's group \(F\). Israel J. Math. 216 (2016), no. 1, 15–59.
• Cannon, J. W.; Floyd, W. J.; Parry, W. R. Introductory notes on Richard Thompson's groups. Enseign. Math. (2) 42 (1996), no. 3–4, 215–256.
• Grigorchuk, R. I.; Nekrashevich, V. V.; Sushchanskiĭ, V. I. Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 134–214; English translation from Proc. Steklov Inst. Math. 2000, no. 4(231), 128–203.
• Hyde, James; Moore, Justin Tatch. Subgroups of \(\mathrm{PL}_+ I\) which do not embed into Thompson's group. arXiv:2103.14911.
• Lehnert, J.; Schweitzer, P. The co-word problem for the Higman-Thompson group is context-free. Bull. Lond. Math. Soc. 39 (2007), no. 2, 235–241.
• Lodha, Yash; Moore, J. T. A nonamenable finitely presented group of piecewise projective homeomorphisms. Groups Geom. Dyn. 10 (2016), no. 1, 177–200.
• McKenzie, Ralph; Thompson, Richard J. An elementary construction of unsolvable word problems in group theory. Word problems: decision problems and the Burnside problem in group theory (Conf., Univ. California, Irvine, Calif. 1969; dedicated to Hanna Neumann), Studies in Logic and the Foundations of Math., 71, pp. 457–478. North-Holland, Amsterdam, 1973.
• Monod, Nicolas Groups of piecewise projective homeomorphisms. Proc. Natl. Acad. Sci. USA 110 (2013), no. 12, 4524–4527.
• Moore, Justin Tatch A brief introduction to amenable equivalence relations. Trends in set theory, 153–163, Contemp. Math., 752, Amer. Math. Soc., Providence, RI, 2020.
• Moore, Justin Tatch Fast growth in the Følner function for Thompson's group \(F\). Groups Geom. Dyn. 7 (2013), no. 3, 633–651.
• Moore, Justin Tatch Hindman's theorem, Ellis's lemma, and Thompson's group \(F\). Zb. Rad. (Beogr.) 17(25) (2015), Selected topics in combinatorial analysis, 171–187.
• Moore, Justin Tatch Nonexistence of idempotent means on free binary systems. Canad. Math. Bull. 62 (2019), no. 3, 577–581.
• Rubin, Matatyahu On the reconstruction of topological spaces from their groups of homeomorphisms. Trans. Amer. Math. Soc. 312 (1989), no. 2, 487–538.
• Salo, Ville Conjugacy of reversible cellular automata. arXiv:2011.07827.