On possible values of the group complexity function of infinite words

Abstract: A classical notion of a factor complexity of an infinite word is defined as a function $p(n)$ counting, for each $n$, the number of distinct factors (or blocks of consecutive letters) of the word of length $n$. The notion has various generalizations and variants. For example, the abelian complexity $p_{ab}(n)$ counts the number of distinct factors of each length $n$ up to abelian equivalence, i.e., only the numbers of occurrences of letters are taken into account, and not their order. The notion of a group complexity generalizes both notions of a factor and an abelian complexities. Namely, given a sequence $ω=(G_n)<em>{n=1}<sup>{\infty}$ of subgroups of the symmetric group $S_n$, the group complexity $p</em>ω(n)$ of a word counts the number of classes of factors of each length $n$ of the word, where words obtained from one another by permutations from $G_n$ are put in the same class. Taking $G_n=S_n$, we obtain the abelian complexity, and taking $G_n=Id$, we recover the factor complexity. Clearly, the group complexity value is between the abelian and the factor complexities. In this paper, we are interested in the following property of words. We say that an infinite word has universal group complexity if for each length $n$ and for each $k$ satisfying $p_s^{ab}(n) \leqslant k \leqslant p_s(n)$, there exists a group $G \in S_n$ such that $p_s^G(n) = k$. In other words, all ``intermediate'' values of complexity can be obtained. We show that Sturmian words satisfy the universal group complexity property, while they are not the only ones. We also study the universal group complexity property for aperiodic ternary words of minimal complexity and for eventually periodic words.

• The paper proves that every Sturmian word realizes every complexity between abelian and factor complexity through a nested chain of Young subgroups.
• The paper establishes a parity-based obstruction for binary words closed under letter exchange, showing in particular that the Thue–Morse word lacks universal group complexity.
• The paper nearly classifies universal group complexity for ternary words of minimal factor complexity and gives a finite algorithmic test for eventually periodic words, while identifying open cases involving complexity 4.

Background and motivation

The factor complexity function $p_w(n)$ of an infinite word $w$, counting distinct factors of length $n$, has been studied since Morse and Hedlund's foundational work on symbolic dynamics. Its abelian counterpart $p^{ab}_w(n)$, which counts factors up to permutation of letters (i.e., up to Parikh vector), was introduced later and admits its own minimality theory: Sturmian words are simultaneously the aperiodic words of minimal factor complexity ($p(n) = n+1$) and minimal abelian complexity ($p^{ab}(n) = 2$). The paper under review works within the framework of group complexity, introduced by Charlier, Puzynina and Zamboni [145.3381–3394, Proc. AMS], which interpolates between these two notions.

Given a sequence $\omega = (G_n)_{n\geq 1}$ with $G_n \leq S_n$, the group complexity $p^\omega_w(n)$ counts equivalence classes of length-$n$ factors under the natural action of $w$0 on positions. Taking $w$1 recovers factor complexity; taking $w$2 recovers abelian complexity. For every subgroup one has

$w$3

The central object of the paper is the universal group complexity property: a word $w$4 has this property if for every $w$5 and every integer $w$6 with $w$7 there exists a subgroup $w$8 such that $w$9. The question is thus an "intermediate values" problem for group complexity, analogous in spirit to inverse problems for factor complexity.

Universal group complexity of Sturmian words

The first main result states that every Sturmian word has universal group complexity. The proof exploits the structure of the lexicographic array of a Sturmian word: consecutive factors differ either by swapping adjacent $n$0 or at the unique right special factor via $n$1. Consequently, the two abelian classes (poor and rich in 1's) appear as contiguous blocks in lexicographic order.

For each $n$2, consider the Young-type subgroup $n$3, which fixes the first $n$4 positions and permutes the rest arbitrarily. An induction on $n$5 shows that passing from $n$6 to $n$7 merges exactly one pair of classes into one, so that $n$8. Since $n$9 ranges over $p^{ab}_w(n)$0, all values from $p^{ab}_w(n)$1 (abelian complexity) through $p^{ab}_w(n)$2 (factor complexity) are attained. This is a clean structural result: the entire interval of possible complexities is realized by a single nested chain of subgroups.

A necessary condition and failure for the Thue–Morse word

Universal group complexity is not characteristic of Sturmian words — certain morphic images of Sturmian words also satisfy it — but it does fail for some prominent words. The authors establish a necessary condition: if a binary word is closed under the exchange morphism $p^{ab}_w(n)$3 and satisfies $p^{ab}_w(n)$4 for some odd $p^{ab}_w(n)$5, then $p^{ab}_w(n)$6 lacks the property. The argument is a parity obstruction: for odd $p^{ab}_w(n)$7, antipodal pairs of factors behave identically under any subgroup action, so group complexity can never take values of the form $p^{ab}_w(n)$8.

As a corollary, the Thue–Morse word does not have universal group complexity. This provides a concrete separation between Sturmian words and other classical low-complexity words.

Ternary words of minimal complexity

Aperiodic words over $p^{ab}_w(n)$9 of minimal complexity $p(n) = n+1$0 were classified by Kaboré and Tapsoba into three types: $p(n) = n+1$1 (a Sturmian word prefixed by a distinguished letter), $p(n) = n+1$2 with $p(n) = n+1$3, and $p(n) = n+1$4 with $p(n) = n+1$5. The paper analyzes each type:

• Type I words have universal group complexity, since their factors are those of a Sturmian word plus one isolated prefix beginning with 2.
• Type II words satisfy the property for every length except $p(n) = n+1$6: at length 2 the factors are $p(n) = n+1$7, giving $p(n) = n+1$8 and $p(n) = n+1$9, but $p^{ab}(n) = 2$0 has only two subgroups, so the value 3 is unattainable. For odd lengths, the construction lifts subgroups acting on odd and even indices separately, using Cartesian products $p^{ab}(n) = 2$1 to realize sums of achievable complexities; for even lengths, abelian complexity drops to 2 and value 3 is obtained via $p^{ab}(n) = 2$2. Notably, if the letter 2 is identified with the least frequent letter of the underlying Sturmian word, the resulting image $p^{ab}(n) = 2$3 recovers universal group complexity for all lengths.
• Type III words are more delicate. Using a new four-rule scheme ($p^{ab}(n) = 2$4, $p^{ab}(n) = 2$5, $p^{ab}(n) = 2$6, $p^{ab}(n) = 2$7) describing consecutive factors in the lexicographic array — proved by an intricate case analysis relying on a density lemma for corresponding Sturmian factors — the authors show that all intermediate values except possibly 4 are achieved. Specifically, the subgroups $p^{ab}(n) = 2$8 yield exactly $p^{ab}(n) = 2$9 for $\omega = (G_n)_{n\geq 1}$0, realizing the range $\omega = (G_n)_{n\geq 1}$1.

The missing value 4 is treated separately. The abelian complexity of a type III word is always 3 or 4, and it equals 4 exactly when rule 3 is applied between rules 2 and 4 in the scheme, which is further characterized combinatorially: $\omega = (G_n)_{n\geq 1}$2 if and only if the last column of the lexicographic array has the form $\omega = (G_n)_{n\geq 1}$3; in particular, this holds whenever the lexicographically smallest factor of length $\omega = (G_n)_{n\geq 1}$4 ends in 0. When the abelian complexity is 3, whether group complexity 4 is attainable depends on the word and the length, and remains unresolved in general. However, the paper proves a strong negative structural result: for infinitely many odd lengths (those derived from convergents $\omega = (G_n)_{n\geq 1}$5 of the slope), if $\omega = (G_n)_{n\geq 1}$6 then $\omega = (G_n)_{n\geq 1}$7 must act transitively on $\omega = (G_n)_{n\geq 1}$8. The proof uses cyclic-complexity techniques: for suitable lengths, the reduced lexicographic array consists of cyclic shifts organized into overlapping "boxes," and non-transitive groups force at least five Parikh-vector classes among the factors. Thus non-transitive subgroups provably cannot fill the gap at these lengths.

Eventually periodic words

For eventually periodic words, the paper gives a complete, algorithmically checkable criterion. If $\omega = (G_n)_{n\geq 1}$9 has period $G_n \leq S_n$0 and $G_n \leq S_n$1 denotes the smallest length at which factor complexity stabilizes, then for $G_n \leq S_n$2 the group complexity relative to any $G_n \leq S_n$3 equals that relative to $G_n \leq S_n$4 in $G_n \leq S_n$5: extending factors by cyclic shifts of the period neither splits nor merges $G_n \leq S_n$6-classes. It follows that universal group complexity for all lengths reduces to a finite check up to length $G_n \leq S_n$7. The word $G_n \leq S_n$8 is exhibited as a periodic example satisfying the property, verified explicitly for lengths up to 7.

Limitations and open questions

Several gaps remain. First, no classification of words with universal group complexity is known; Sturmian words satisfy it but do not exhaust the class, as shown by type I ternary words and images like $G_n \leq S_n$9. Second, the authors propose studying an eventual version of the property (holding from some length onward), which strictly enlarges the class — type II words belong to it but not to the all-lengths version. Third, Open problem 3 asks precisely when group complexity 4 is attained for type III words at lengths with abelian complexity 3; the transitivity theorem restricts candidate subgroups but does not settle existence. Finally, the authors conjecture that the ternary classification extends to larger alphabets, while noting the expected increase in technical difficulty. One should also note that the necessary condition excluding the Thue–Morse word relies on closure under letter exchange, so it applies only to a restricted family of binary words.

Conclusion

This paper initiates the systematic study of which intermediate values between abelian and factor complexity are realizable by permutation-group quotients. Its principal contributions are: the positive result that Sturmian words achieve every intermediate value via a nested chain of Young subgroups; a parity-based obstruction showing the Thue–Morse word fails the property; a near-complete analysis of ternary minimal-complexity words, including a new lexicographic scheme for type III words and a transitivity constraint on subgroups attaining complexity 4; and a finite algorithmic criterion for eventually periodic words. The work frames a tractable intermediate-values question that connects combinatorics on words with permutation group theory, and leaves open the full classification of words with universal group complexity.