# Key Results of Combinatorial Group Theory.

What are the key theorems of combinatorial group theory?

By "key theorems", I mean those most commonly used in the literature.

For added context, I have copies of "Presentation of Groups," by D. L. Johnson and "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al., and I've just started a Ph.D. in the area.

I suppose a good place to start would be

Theorem (Nielson-Shreier Theorem): Every subgroup of a free group is itself free.

Update: I've got a copy of Lyndon & Schupp's "Combinatorial Group Theory".

• You should pick up the Lyndon-Schupp book too. And Cohen's book. – Steve D Dec 7 '17 at 19:46
• en.wikipedia.org/wiki/Freiheitssatz – Steve D Dec 7 '17 at 19:46
• – Dietrich Burde Dec 7 '17 at 19:54
• At this point, do not worry about "key results" (there are way too many), worry about understanding the basic definitions and constructions. – Moishe Kohan Dec 7 '17 at 21:32
• The Reidemeister-Schreier theorem seems important enough to mention here. – Shaun Dec 14 '17 at 18:12

My understanding of combinatorial group theory came as the result of a personal relationship between my advisor G. Baumslag and W. Magnus who ran a joint (CUNY & NYU) group theory seminar in New York in the 60s & 70s. The origin story that we grad students heard was that Max Dehn (see Wikipedia) started CGT when he posed the Word Problem, the Conjugacy Problem, and the Isomorphism Problem for Group Presentations. These of course had topological roots but some of us ignored that at our peril.
Some of the main results of CGT from my study in those days, that I can remember, is the solution of Dehn's three problems for free groups, the first two for all, and special cases of the third, for one-relator groups, solutions of problems for knot and braid groups, and understanding the structure of free products and amalgamated free products (the word problem modulo that of the factors). CGT was of course also important in proofs of unsolvability.
I am not up on the main results after the mid 70s.

• Yes, things have changed quite a bit since 1970s. – Moishe Kohan Dec 8 '17 at 1:05
• @MoisheCohen Things have changed, yes, but not so much - knot groups, braid groups, graphs of groups (generalisations of free products with amalgamation) - these are all still key areas. Probably the biggest difference is that there is a greater emphasis on geometric and topological methods in the post-Gromov era. – user1729 Dec 8 '17 at 20:29
• I agree with the 2nd sentence and disagree with the 1st (except for Braid groups). Think of somebody who can nowdays say that their main focus of research is graphs of groups. In contrast, think of those who can say that their main focus of research is cube complexes. (One can say, of course, that graphs of groups remain as one of the key group theoretic constructions.) Now, think of somebody whose main focus are knot groups. (The latter became a special case of 3-manifold groups and these are still intensively studied.) – Moishe Kohan Dec 8 '17 at 20:44
• @MoisheCohen I was particularly thinking of JSJ-decompositions when I wrote the bit about graphs of groups - very roughly, these are graphs of groups where every edge group is virtually-$\mathbb{Z}$. They underlie the isomorphism problem for hyperbolic groups, the Sela/Kharlampovich-Myasnikov stuff on Tarski's problem, etc. – user1729 Dec 8 '17 at 20:49
• Also, I am pretty sure that the conjugacy problem for one-relator groups is still open. The only claimed proof I could find was a "sketched" proof from '92, but it was apparently still open in the early 2000s. (Also, I figure this paper implies its still open, going by the parenthesis in the abstract, but my department doesn't subscribe to Comm. Algebra for some reason.) – user1729 Dec 8 '17 at 20:51

I'm not sure where are boundaries between combinatorial group theory, low-dimensional topology, group homology and geometric group theory (and whether they exist), but here's some list. It's partly not "results", but constructions, but I think that constructions are more key-ish than theorems. (Also I have heavy bias towards homological and homotopical aspects of CGT.)

• Lyndon 4-term partial resolvent of trivial module built from presentation, relation modules and $$\pi_2$$ of presentation complex (which is faithful $$\Bbb Z[G]$$-module)

• (Higher) Magnus embedding and Fox free differential calculus

• Commutator calculus, Hall sets and Zassenhaus series

• Weighted deficiency and infinite Golod groups

• Ol'shanski theory of relative small cancellation

• Various theorems about ends of groups, splittings as a graph of groups and relation with deficiency and Betti numbers

• (already mentioned) Freiheitssatz and generalizations for presentations in a variety (Romanovskii, C. Gupta)

• Nikolov-Segal theorem, word mappings and word width

• Birkhoff's HSP theorem and its quasivariety generalization

• Diamond lemma

• particular Stallings theorem (if $$q: G \to H$$ is iso on abelianisation and epi on $$H_2(-, \Bbb Z)$$, then $$q$$ gives isomorphism on factors by LCS)

• $$F/[R, R]$$ is torsion-free (and generalizations of this by Passi, Stohr et al)

• Thank you, @xsnl. I'll work my way through these and, if possible, try to understand them. – Shaun Dec 1 '18 at 3:56