Year: 2021
Type: preprint
Abstract: We present efficient algorithms to decide whether two given counting functions on non-abelian free groups or monoids are at bounded distance from each other and to decide whether two given counting quasimorphisms on non-abelian free groups are cohomologous. We work in the multi-tape Turing machine model with non-constant time arithmetic operations. In the case of integer coefficients we construct an algorithm of linear space and time complexity (assuming that the rank is at least $3$ in the monoid case). In the case of rational coefficients we prove that the time complexity is $O(N\log N)$, where $N$ denotes the size of the input, i.e. it is as fast as addition of rational numbers (implemented using the Harvey--van der Hoeven algorithm for integer multiplication). These algorithms are based on our previous work which characterizes bounded counting functions.
Source: arXiv (Cornell University)
Authors: Hartnick, Tobias, Talambutsa, Alexey
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4387224687