Growth and Language Complexity of Potentially Positive Elements of Free Groups

A word in a free group is called ``potentially positive’’ if it is automorphic to an element which is written with only positive exponents. We will develop automata to analyze properties of potentially positive words. We will use these to give new bounds on the asymptotic growth of potentially positive elements in free groups of 2 to 7 generators. We prove the bounds for $F_2$ are tight, giving the growth function up to a constant multiplier. We use the same tools to show that certain restricted automata cannot recognize the set of potentially positive elements.

💡 Research Summary

The paper investigates the combinatorial growth and language‑theoretic complexity of a class of words in free groups called “potentially positive” (PP). A word w in the free group (F_r=\langle x_1,\dots ,x_r\rangle) is called potentially positive if there exists an automorphism (\varphi) of the group such that (\varphi(w)) can be written using only positive exponents of the generators. This notion generalizes positive words (which contain no inverses) and captures the idea that many algebraic properties of positive one‑relator groups are preserved under automorphisms.

The central question (Question 1.1) asks: how many potentially positive words of length n exist in (F_r)? The authors answer this completely for rank 2 and give improved lower bounds for ranks 3 through 7.

Methodology – Automata and Spectral Analysis
The authors develop a framework based on deterministic finite automata (treated as labelled directed graphs) that generate only PP words. They impose three crucial properties on an automaton A:

1. Reduced – no edge connects a state to the inverse label of another, guaranteeing that any path spells a reduced word.
2. Mixing – for some N, every ordered pair of states can be connected by a path of any length ≥ N. This ensures strong connectivity and allows the use of symbolic‑dynamics results.
3. One‑to‑constant labeling – each cyclically reduced word is produced by at most a constant number c of closed paths in A. Lemma 3.5 shows that this holds precisely when distinct closed paths never share the same start, end, and label.

For a mixing automaton, Lemma 3.2 (and Corollary 3.3) states that the number of closed paths of length n grows like (\Theta(\lambda^n)), where (\lambda) is the largest eigenvalue of the adjacency matrix. If the labeling is one‑to‑constant, the same growth rate transfers to the number of distinct words produced (Lemma 3.6). Thus, constructing a suitable automaton reduces the growth problem to a spectral computation.

Rank‑2 Result – Exact Exponential Growth
Figure 4.1 (described in the text) presents a concrete automaton for (F_2). It starts from the automaton that generates ordinary positive words and repeatedly applies a family of automorphisms that “twist’’ the generators, thereby enlarging the state space and increasing the spectral radius. The resulting adjacency matrix has characteristic polynomial \