Logspace and compressed-word computations in nilpotent groups

For finitely generated nilpotent groups, we employ Mal’cev coordinates to solve several classical algorithmic problems efficiently. Computation of normal forms, the membership problem, the conjugacy problem, and computation of presentations for subgroups are solved using only logarithmic space and quasilinear time. Logarithmic space presentation-uniform versions of these algorithms are provided. Compressed-word versions of the same problems, in which each input word is provided as a straight-line program, are solved in polynomial time.

💡 Research Summary

The paper investigates the computational complexity of several fundamental algorithmic problems in finitely generated nilpotent groups, focusing on two modern complexity measures: logarithmic‑space (log‑space) computation and compressed‑word (straight‑line program) computation. Using Mal’cev (Hall‑Mal’cev) bases, the authors represent each group element by an integer coordinate vector. They show that group multiplication and exponentiation correspond to polynomial functions on these coordinates, and crucially, the size of the coordinates of a product of n elements grows only polynomially in n, with the degree depending solely on the nilpotency class (a fixed constant).

The paper first formalises log‑space computation via log‑space transducers and defines compressed words as straight‑line programs in Chomsky normal form. It then develops a suite of algorithms:

1. Normal form computation (word problem) – given a word, compute its Mal’cev normal form. The algorithm runs in O(log L) space and O(L log³ L) time (even O(L log² L) for certain subroutines).

2. Membership problem – decide whether a given element belongs to a subgroup generated by a fixed set, and if so, express it as a product of the generators. This uses a matrix‑reduction (Gaussian elimination on coordinate matrices) performed in log‑space.

3. Subgroup presentation – from the reduced matrix obtain a finite presentation for the subgroup, again within the same space‑time bounds.

4. Homomorphism kernel and pre‑image – given a homomorphism specified on generators, compute a generating set for its kernel and find a pre‑image of a given element.

5. Centralizers and conjugacy – compute a generating set for the centralizer of an element and decide conjugacy of two elements, producing a conjugating element when it exists.

All the above algorithms run in logarithmic space and quasilinear time (O(L log³ L) overall, with some sub‑problems in O(L log² L)). When the number of generators and the nilpotency class are bounded, the algorithms are uniform: the group presentation may be part of the input without affecting the asymptotic bounds.

For the compressed‑word setting, each input word is given as a straight‑line program of size L, which can encode words of exponential length. The authors first compute binary Mal’cev coordinates directly from the program, then apply the log‑space algorithms to the coordinate vectors. This yields polynomial‑time solutions: all six problems are solved in O(L⁴) time, with normal form, centralizer, and conjugacy in O(L³) time.

The paper also discusses related work, contrasting the Mal’cev‑basis approach with linear‑group embeddings and previous decidability results, and highlights that prior literature lacked concrete complexity estimates for nilpotent groups. By providing explicit log‑space and polynomial‑time bounds, the authors fill this gap and demonstrate that many classical group‑theoretic decision problems become practically feasible even for compressed inputs.

Overall, the work establishes that finitely generated nilpotent groups admit log‑space computable normal forms, and that all standard algorithmic problems (membership, conjugacy, kernel, presentations) can be solved efficiently both in the ordinary and compressed settings. This advances the state of the art in algorithmic group theory and opens avenues for applications in areas such as automated theorem proving, cryptographic protocol analysis, and the study of automorphism groups of nilpotent structures.