This is just a reference request. I'm trying to find out whether there are some well developed notes/theory out there (books and the like) focusing on cyclically reduced words in groups. Quickly searching Google, all I could find was Wikipedia and this: Cyclically reduced words in free groups

Any more bibliography would be deeply appreciated.

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2 Answers2


I would be somewhat surprised if notes that focus on cyclically reduced words exists, as I doubt they are interesting to that degree, and I would be surprised if there was much to say. (Maybe some notes outlining the basics are around, from a class or something)

Both Combinatorial Group Theory books (the ones by Lyndon and Schupp, and the other by Magnus, Karrass, and Solitar will use the concept of cyclically reduced to solve conjugacy problem in free groups and related groups, and in doing that they figure out the basic properties (like cyclically reduced words which are conjugate have the same length). I actually think normal form is probably the more important concept.

When doing small cancellation theory it is typical to assume your relations in your presentation are cyclically reduced, mostly out of convenience. The classic source of small cancellation theory is Combinatorial Group Theory by Lyndon and Schupp. Another source to learn small cancellation theory is Geometry of Defining Relations in Groups by Ol'shanksii


You may be interested in "van Kampen diagrams". I once wrote out a math.SE answer about them here. Basically, diagrams over a presentation $\mathcal{P}$ are "cyclic" words - words written on a circle equal to the trivial word in $\mathcal{P}$. The boundary of a diagram is a cyclically reduced word. (My references are the same as Paul Plummers last two, as they are completely standard: Combinatorial Group Theory by Lyndon and Schupp and Geometry of Defining Relations in Groups by Ol'shanksii.)

Diagrams form the central idea of small cancellation theory, as mentioned in Paul Plummer's answer. The small cancellation conditions relate to tilings by diagrams. But the theory of diagrams does not stop here!

Another nice aspect of diagrams is asphericity: diagrams are 2-dimensional objects so you can try and tile a sphere with them, and a presentation $\mathcal{P}$ is aspherical if every tiling of the sphere by diagrams over $\mathcal{P}$ contains an inverse-pair of diagrams. (This inverse-pair then cancel, so you have a "smaller" tiling, which again contains an inverse-pair, which cancel, etc. until you have nothing left. So no "solid" tilings!) This is related to the standard notion of topological asphericity: the reference is IM Chiswell, DJ Collins, J Huebschmann, Aspherical group presentations Mathematische Zeitschrift 1981.

As another neat example, diagrams lend themselves to a notion of "area" in groups. This is formalised using something called a Dehn function. In hyperbolic space, area is linear: think $\pi r$ for area of a circle rather than $\pi r^2$. So we can define hyperbolic groups to be precisely those groups with linear Dehn function. This definition is equivalent to the more common definition of a $\delta$-hyperbolic group, due to Gromov (and the equivalence was proven by Gromov in his original article).

(As an aside, my PhD supervisior used to always start thinking about a problem on words by writing them on a circle, and diagrams are a sort of special-case of this. For example, "spelling theorems", such as the Newman-Gurevich spelling theorem for one-relator groups, become much clearer when stated about cyclic words.

In particular, the Newman-Gurevich spelling theorem is as follows. Suppose $G=\langle X; R^n\rangle$ with $n\geq1$, then a \emph{Gurevich subword for $R^n$} of a word $W$ is a subword of $W$ which has the form $S^{n-1}S_0$ where $S=S_0S_1$ is a cyclic shift of $R$ or $R^{-1}$, and every generator which appears in $R$ appears in $S_0$.

Theorem. Let $G=\langle X; R^n\rangle$, $n\geq1$. Suppose $W=_G 1$ but $W$ is freely reduced and not the empty word. Then $W$ contains a Gurevich subword for $R^n$. If, further, $W$ is cyclically reduced, then either $W$ is a cyclic shift of $R^n$ or $R^{-n}$, or some cyclic shift of $W$ contains two disjoint subwords, each of which is a Gurevich subword for $R^n$.

Re-stated in terms of cyclic words, it simply says that the cyclic word $W$ contains two disjoint Gurevich subwords or is $R^{\pm n}$.)

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