# TC^0 circuits for algorithmic problems in nilpotent groups

## TL;DR

This paper demonstrates that key algorithmic problems in finitely generated nilpotent groups are complete for the circuit class TC^0, extending previous Logspace results and showing their computational efficiency within this class.

## Contribution

The paper proves that multiple algorithmic problems in nilpotent groups are TC^0-complete, and establishes the TC^0 complexity of the unary extended gcd problem, broadening understanding of their computational complexity.

## Key findings

- Problems are TC^0-complete for finitely generated nilpotent groups.
- Unary extended gcd problem is in TC^0.
- Word problem and normal form computations are in uniform TC^0 with binary inputs.

## Abstract

Recently, Macdonald et. al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in Logspace. Here we follow their approach and show that all these problems are complete for the uniform circuit class TC^0 - uniformly for all r-generated nilpotent groups of class at most c for fixed r and c. In order to solve these problems in TC^0, we show that the unary version of the extended gcd problem (compute greatest common divisors and express them as linear combinations) is in TC^0. Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform TC^0, while all the other problems we examine are shown to be TC^0-Turing reducible to the binary extended gcd problem.