Decision Problems in Group Theory (L24)
Non-Examinable (Part III Level)
Maurice Chiodo
Overview
The aim of this course is to investigate incomputable problems in group theory. We will define several different methods of computation and use these to show various problems to be
incomputable, as well as determine the degrees of incomputability of such problems.
Course description
Computability theory: Turing machines, recursive and recursively enumerable sets, the halting
set K. Church’s thesis. Many-one and Turing reductions, Kleene hierarchy. Modular machines
and their equivalence to Turing machines. Minsky machines and how they simulate Turing
machines. First order theories.
Group theory: Recursive presentations of groups, the word and isomorphism problems with
basic properties and examples. Basic computable properties of groups. Post’s construction of
a finitely presented semigroup with unsolvable word problem.
Embedding theorems: A finitely presented group with unsolvable word problem; Higman’s embedding theorem. A universal finitely presented group. The Adian-Rabin construction, unrecognisability of Markov properties. The Boone–Higman theorem. Properties preserved by
Higman embeddings. Degrees of various incomputable problems.
Finite quotients: Slobodskoı̆’s theorem on undecidability of the first order theory of finite groups.
Bridson–Wilton theorem on undecidability of finite quotients.
Pre-requisites
It will be assumed that you have attended a first course in group theory, and that you attend
at least the first 4 lectures of Part III Geometric Group Theory (Lent). In addition, Part II
Automata and Formal Languages (Michaelmas) and/or Part III Logic (Lent) would be helpful
for some intuition in computability theory, but are not essential.
Literature
1. R.I. Soare, Recursively enumerable sets and degrees: a study of computable functions and
computably generated sets. Springer-Verlag (Perspectives in mathematical logic), 1987.
2. C. F. Miller III, Decision Problems For Groups-Survey and Reflections. Algorithms and
classification in combinatorial group theory (Berkeley, CA, 1989), Math. Sci. Res. Inst.
Publ., 23, Springer, New York, 1–59 (1992). Also available at
http://www.ms.unimelb.edu.au/~cfm/papers/paperpdfs/msri_survey.all.pdf
3. J. Rotman, An Introduction To The Theory Of Groups. (GTM 148), Springer, fourth
edition, 1995.
4. D. E. Cohen, Combinatorial Group Theory: A Topological Approach. London Mathematical Society Student Texts, Cambridge University Press, 1989.
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5. A. M. Slobodskoı̆, Undecidability of the universal theory of finite groups (Russian), Algebra
i Logika 20, no. 2, 207–230 (1981). English transl., Algebra and Logic 20, no. 2, 139–156
(1981).
6. M. Bridson, H. Wilton, The triviality problem for profinite completions, Invent. Math.
202, no. 2, 839–874 (2015).
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