Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.14536

Twisted Equations and Palindrome Length

Volker Diekert·Anna E. Frid·Silas Natterer·Alexander Thumm
View original source ↗Read source document ↗Export metadata ↓

Abstract

Twisted equations in free groups are equations in which a finite group of symmetries acts on the free group by automorphisms and anti-automorphisms, such as inversion and reversal; moreover, the variables may be subject to rational constraints. Algorithms from the theory of equations in virtually free groups solve such equations in PSPACE, but they are intricate. Our main result is an elementary algorithm for quadratic twisted equations, which extends the algorithm of Comerford and Edmunds (1981) for quadratic equations in free groups to the twisted setting and to rational constraints. Besides deciding solvability in PSPACE and yielding EDT0L descriptions of solution sets, our algorithm computes, for every quadratic expression, finitely many parametric forms describing its values as freely reduced words. As an application, we study the palindrome length of elements of free groups, that is, the least number of palindromes whose product they are. Guided by the parametric forms that our algorithm computes for products of palindromes, we found, and then proved directly, a characterization of the freely reduced words of palindrome length at most $n$ in terms of patterns describing how palindromes are nested by twisted conjugations, thereby confirming a Catalan structure conjectured by the second author. Consequently, these words form a context-free language, and palindrome length is computable in cubic time as well as in LOGCFL, answering a question of Bardakov, Shpilrain, and Tolstykh (2005).

Explore related subjects

math.GRcs.FL

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Volker Diekert, Anna E. Frid, Silas Natterer, Alexander Thumm. 2026-10-10. Twisted Equations and Palindrome Length. https://arxiv.org/abs/2610.14536

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Almost tight asymptotic enumeration of finite $p$-groups: Almost all finite groups are $2$-groups of class $2$

Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. We prove, uniformly over all primes $p$, that \[ \log_p f(p^m)=\frac{2}{27}m^3-\frac49m^2+O(m), \] with an absolute implied constant. This improves the classical upper bounds of Sims (\emph{Proc. Lond. Math. Soc.}, 1965) and Newman--Seeley and matches Higman's lower bound (\emph{Proc. Lond. Math. Soc.}, 1960) through the quadratic term. We also prove that \emph{the proportion of $2$-groups of nilpotency class $2$ among all groups of order at most $n$, counted up to isomorphism, tends to $1$ as $n\to\infty$.} These results answer questions~22.3 and~22.19 of Blackburn--Neumann--Venkataraman (\emph{Enumeration of Finite Groups}, 2007). Our proof refines Sims' enumeration framework using graded Lie brackets, constraints on deeper commutator coordinates, and incidence and Fourier estimates for alternating forms. Normalization by changes of generating bases then determines the quadratic term in the upper bound. We also prove that all but a $p^{-Ω(m^2)}$ fraction of groups of order $p^m$ have a central elementary abelian Frattini subgroup. Combining this concentration with the sharp enumeration bound of Bläser--Li--Qiao--Rogovskyy (\emph{arXiv}, 2025) for $p$-groups of Frattini class $2$ yields the linear error term. The same presentation estimates give fixed-quotient extension bounds which, together with Pyber's prescribed-Sylow theorem and the concentration result, establish domination by $2$-groups of nilpotency class $2$.

math.GR↗
arXiv2026

Algebro-Geometric Invariants of Finitely Generated Groups (The Profile of a Representation Variety)

If G is a finitely generated group, and A an algebraic group, then Hom(G,A) is a possibly reducible algebraic variety denoted by R_A(G). Here we define the profile function, P_d(R_A(G)), of the representation variety of G over A to be P_d(R_A(G))=(N_d(R_A(G)),...,N_0(R_A(G))), where N_i(R_A(G)) stands for the number of irreducible components of R_A(G) of dimension i, where 0\leq i\leq d, and d=Dim(R_A(G)). We then use this invariant in the study of fg groups and prove various results. In particular, we show that if G an orientable surface group of genus g\geq 1, then P_d(R_{SL(2,C)}(G))\neq P_d(R_{PSL(2,C)}(G)). We also show that the same holds for G a torus knot group with presentation where both p,t are greater than 2, and that the same also holds when G is a the fundamental group of a compact non-orientable surface of genus g\geq 3. Further, we show that if a group G can be n+1 generated, and presented by , where W is a non-trivial word in F_n= , and A=PSL(2, C), that then Dim(R_{A}(G)) is equal to Max{3n, Dim(R_{A}(G'))+2 \} \leq 3n+1, where G'= . We also give a condition guaranteeing that the resulting algebraic variety is reducible.

math.GR↗
arXiv2026

Exact versus monotone isoperimetric profiles of finitely generated groups

Gromov asked whether the exact isoperimetric profile of an infinite finitely generated group is comparable, at the same cardinality, to its greatest nondecreasing minorant. We compare finite sets with their translates and construct orders for which the first $r$ vertices have boundary bounded by a fixed multiple of that of the whole set. Such orders follow from profile summability or from rays in Schreier graphs with a uniform bound on the number using each edge. Among infinite finitely generated groups, we obtain comparison and nested exhaustions for elementary amenable and branch groups, direct products of at least two infinite groups, restricted wreath products with two nontrivial factors, and certain simple topological full groups, including amenable derived groups of interval exchanges. The exhausting $n$-point sets have boundary comparable to the monotone profile at $n$. The ray condition is quasi-isometry invariant and holds for the Sierpiński-carpet group. For polycyclic groups of exponential growth, we refine Pittet's construction to obtain explicit nested sets with $|F_n|=n$ and $|\partial F_n|\asymp n/\log n$, answering his question with exact rather than comparable cardinalities. We also prove that the normalized $L^1$ profile satisfies $j(v)\le C(v/u)j(u)$ for $1\le u\le v$ on every infinite connected bounded-degree vertex-transitive graph. This yields the separation bound asked for by Gournay--Le Coz and Hume's Dirichlet--Neumann comparison at $p=1$. For groups of exponential growth, the exact-to-monotone comparison ratio is $O(\log\log n)$. We answer the first part of Correia's Question 1.3 positively by proving the required dilation regularity, and the second negatively by constructing elementary amenable examples whose inverse compositions fail this regularity for every choice of equivalent representatives.

math.GR↗