cyclically reduced

Let $M(X)$ be a free monoid with involution ${}^{-1}$ on $X$. A word $w\in M(X)$ is said to be cyclically reduced if every cyclic conjugate of it is reduced. In other words, $w$ is cyclically reduced iff $w$ is a reduced word and that if $w=uv{u}^{-1}$ for some words $u$ and $v$, then $w=v$.

For example, if $X=\{a,b,c\}$, then words such as

$$c^{-1}b{c}^{2}a\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}aba{c}^{2}b{a}^2$$

are cyclically reduced, where as words

$$a^{2}bc{a}^{-1}\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}c{b}^2{b}^{3}c$$

are not, the former is reduced, but of the form $a(abc)a^{-1}$, while the later is not even a reduced word.

Remarks. The concept of cyclically reduced words carries over to words in groups. We consider words in a group $G$.

• If a word is cyclically reduced, so is its inverse and all of its cyclic conjugates.

• A word $v$ is a cyclic reduction of a word $w$ if $w=uv{u}^{-1}$ for some word $u$, and $v$ is cyclically reduced. Clearly, every word and its cyclic reduction are conjugates of each other. Furthermore, any word has a unique cyclic reduction.

• Every group $G$ has a presentation $⟨S|R⟩$ such that

  1. $R$ is cyclically reduced (meaning every element of $R$ is cyclically reduced),

  2. closed under inverses (meaning if $u\in R$, then $u^{-1}\in R$), and

  3. closed under cyclic conjugation (meaning any cyclic conjugate of an element in $R$ is in $R$).

  Furthermore, if $G$ is finitely presented, $R$ above can be chosen to be finite.

  Proof.

  Every group $G$ has a presentation $⟨S|R^{\prime }⟩$. There is an isomorphism from $F(S)/N(R^{\prime })$ to $G$, where $F(S)$ is the free group freely generated by $S$, and $N(R^{\prime })$ is the normalizer of the subset $R^{\prime }\subseteq F(S)$ in $F(S)$. Let $R^{\prime \prime }$ be the set of all cyclic reductions of words in $R^{\prime }$. Then $N(R^{\prime \prime })=N(R^{\prime })$, since any word not cyclically reduced in $R^{\prime }$ is conjugate to its cyclic reduction in $R^{\prime \prime }$, and hence in $N(R^{\prime \prime })$. Next, for each $u\in R^{\prime \prime }$, toss in its inverse and all of its cyclic conjugates. The resulting set $R$ is still cyclically reduced. Furthermore, $R$ satisfies the remaining conditions above. Finally, $N(R)=N(R^{\prime \prime })$, as any cyclic conjugate $v$ of a word $w$ is clearly a conjugate of $w$. Therefore, $G$ has presentation $⟨S|R⟩$.

  If $G$ is finitely presented, then $S$ and $R^{\prime }$ above can be chosen to be finite sets. Therefore, $R^{\prime \prime }$ and $R$ are both finite. $R$ is finite because the number of cyclic conjugates of a word is at most the length of the word, and hence finite.∎