Functions on groups and computational complexity

On the topological complexity of toral relatively hyperbolic groups

We prove that the topological complexity $\mathrm{TC}(π)$ equals $\mathrm{cd}(π\timesπ)$ for certain toral relatively hyperbolic groups $π$.

Decomposition complexity growth of finitely generated groups

Finite decomposition complexity and asymptotic dimension growth are two generalizations of M. Gromov's asymptotic dimension which can be used to prove property A for large classes of finitely generated groups of infinite asymptotic dimension. In this paper, we introduce the notion of decomposition complexity growth, which is a quasi-isometry invariant generalizing both finite decomposition complexity and dimension growth. We show that subexponential decomposition complexity growth implies property A, and is preserved by certain group and metric constructions.

群中算法问题的线性平均情况复杂度

Linear average-case complexity of algorithmic problems in groups

本文研究多类群中字问题的平均情况复杂度，证明其通常为线性，并改进矩阵群最坏情况复杂度上界，同时首次讨论恒等问题复杂度。

Topological complexity of basis-conjugating automorphism groups

We compute the topological complexity of Eilenberg-Mac Lane spaces associated to the group of automorphisms of a finitely generated free group which act by conjugation on a given basis, and to certain subgroups.

The Karoubi-Weibel complexity for groups

Let $G$ be a finitely presented group. A new complexity called \textit{Karoubi-Weibel complexity} or \textit{covering type}, is defined for $G$. The construction is inspired by recent work of Karoubi and Weibel \cite{KW}, initially applied to topological spaces. We introduce a similar notion in combinatorial form in order to apply it to finitely presentable groups. Some properties of this complexity as well as a few examples of calculation/estimation for certain classes of finitely presentable groups are considered. Finally we give a few applications of complexity to some geometric problems, n…

Generic-case complexity, decision problems in group theory and random walks

We give a precise definition of ``generic-case complexity'' and show that for a very large class of finitely generated groups the classical decision problems of group theory - the word, conjugacy and membership problems - all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs.

Parametrised topological complexity of group epimorphisms

We show that the parametrised topological complexity of Cohen, Farber and Weinberger gives an invariant of group epimorphisms. We extend various bounds for the topological complexity of groups to obtain bounds for the parametrised topological complexity of epimorphisms. Several applications are given, including an alternative computation of the parametrised topological complexity of the planar Fadell--Neuwirth fibrations which avoids calculations involving cup products. We also prove a homotopy invariance result for parametrised topological complexity of fibrations over different bases.

Descriptive complexity of subsets of the space of finitely generated groups

In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups, groups of exponential growth and groups with decidable word problem are $\mathbfΣ^0_2$-complete and that the sets of periodic groups and groups of intermediate growth are $\mathbfΠ^0_2$-complete. We also provide bounds for the descriptive complexity of simplicity, amenability, residually finiteness, Hopficity and co-Hopficity. This paper is intended to serve as…

An Effective Lower Bound for Group Complexity of Finite Semigroups and Automata

The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, \textit{Complexity of finite semigroups}, Annals of Mathematics (2) \textbf{88} (1968), 128--160, motivated by the Prime Decomposition Theorem of K. Krohn and J. Rhodes, \textit{Algebraic theory of machines, {I}: {P}rime decomposition theorem for finite semigroups and machines}, Transactions of the American Mathematical Society \textbf{116} (1965), 450--464. Here we provide an effective lower bound for group complexity.

寻找给定阶子群的计算复杂性

Computational Complexity of Finding Subgroups of a Given Order

本文研究了在凯莱表表示下寻找给定阶子群的计算复杂性，并为阿贝尔群提出了最优算法。

关于任意有限群上广义隐藏移位问题的样本复杂度

On the sample complexity of the generalized hidden shift problem over arbitrary finite groups

本文提出任意有限群上的广义隐藏移位问题，并基于群论数据给出样本复杂度的上下界。

Average-case complexity and decision problems in group theory

We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on ``generic-case complexity'' we show that if a finitely generated group $G$ has the word problem solvable in subexponential time and has a subgroup of finite index which possesses a non-elementary word-hyperbolic quotient group, then the average-case complexity of the word problem for $G$ is linear time, uniformly with respect to the collection of all length-invariant measures on $G$. For example, the result applies to all bra…

Algorithmic problems in right-angled Artin groups: complexity and applications

In this paper we consider several classical and novel algorithmic problems for right-angled Artin groups, some of which are closely related to graph theoretic problems, and study their computational complexity. We study these problems with a view towards applications to cryptography.