Mark Sapir
This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme properties (monsters).
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We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an algorithm that, given an arbitrary finite presentation of an automatic group $Γ$, will construct explicit finite models for the skeleta of $K(Γ,1)$ and hence compute the integral homology and cohomology of $Γ$.
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.
The theory of Engel groups plays an important role in group theory since these groups are closely related to the Burnside problems. In this survey we consider several classical and novel algorithmic problems for Engel groups and propose several open problems. We study these problems with a view towards applications to cryptography.
We begin with a review of the notion of a braid group. We then discuss some known solutions to decision problems in braid groups. We then move on to proving new results in braid group algorithmics. We offer a quick solution to the generalized word problem in braid groups, in the special case of cyclic subgroups. We illustrate this solution and its complexity using a multitape Turing machine. We then turn to a discussion of decision problems in cyclic amalgamations of groups. Again using a multitape Turing machine, we solve the word problem for the cyclic amalgamation of two braid groups. We an…
We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.
Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ the generalized Stallings' methods, developed by the author, to solve various algorithmic problems concerning finitely generated subgroups of amalgams of finite groups.
Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. It turns out that Stallings' methods can be effectively generalized for the class of amalgams of finite groups. In the present paper we employ subgroup graphs constructed by the generalized Stallings' folding algorithm to solve various algorithmic problems in amalgams of finite groups.
We study direct products of free-abelian and free groups with special emphasis on algorithmic problems. After giving natural extensions of standard notions into that family, we find an explicit expression for an arbitrary endomorphism of $\ZZ^m \times F_n$. These tools are used to solve several algorithmic and decision problems for $\ZZ^m \times F_n $: the membership problem, the isomorphism problem, the finite index problem, the subgroup and coset intersection problems, the fixed point problem, and the Whitehead problem.
A method of random search based on Kolmogorov complexity is proposed and applied to two search problems in group theory. The method is provably effective but not practical, so the applications involve heuristic approximations. Perhaps surprisingly, these approximations seem to work. Some experimental evidence is presented.
We design new deterministic and randomized algorithms for computational problems in free solvable groups. In particular, we prove that the word problem and the power problem can be solved in quasi-linear time and the conjugacy problem can be solved in quasi-quartic time by Monte Carlo type algorithms.
In this paper we further study properties of Magnus embedding, give a precise reducibility diagram for Dehn problems in groups of the form $F/N^{(d)}$, and provide a detailed answer to Problem 12.98 in Kourovka notebook. We also show that most of the reductions are polynomial time reductions and can be used in practical computation.
We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller's groups. We show that for "almost all", in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is worth noting that the Conjugacy Problem in a Miller group may have be undecidable. Our results show that "hard" instances of the problem comprise a negligibly small part of the group.
The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. Epstein showed that geometrically finite hyperbolic groups are biautomatic. Neumann/Reeves showed that virtually central extensions of word hyperbolic groups are biautomatic. We prove the following generalisation: Theorem. Let H be a geometrically finite hyperbolic group. Let sigma in H^2(H) and suppose that sigma restricted to P is zero for any parabolic subgroup P of H. Then the extension of H by sigma is biautomatic. We also prove another generalisation of the result of Epstein. Theorem. Le…
We construct and study finitely presented groups with quadratic Dehn function (QD-groups) and present the following applications of the method developed in our recent papers. (1) The isomorphism problem is undecidable in the class of QD-groups. (2) For every recursive function $f$, there is a QD-group $G$ containing a finitely presented subgroup $H$ whose Dehn function grows faster than $f$. (3) There exists a group with undecidable conjugacy problem but decidable power conjugacy problem; this group is QD.
The main achievement of this thesis is an algorithm which given a finite group presentation and natural numbers n and k, computes all the relators of length and area up to n and k respectively. The complexity of this algorithm is better by a factor which is over-exponential than that of classical methods using van Kampen diagrams.
We show that every countable group H with solvable word problem (=computable group) can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We also give estimates of time and space complexity of the word problem in G and of the membership problem for H < G.
We describe the main questions connected to torsion subgroups in the unit group of integral group rings of finite groups and algorithmic methods to attack these questions. We then prove the Zassenhaus Conjecture for Amitsur groups and prove that any normalized torsion subgroup in the unit group of an integral group of a Frobenius complement is isomorphic to a subgroup of the group base. Moreover we study the orders of torsion units in integral group rings of finite almost quasisimple groups and the existence of torsion-free normal subgroups of finite index in the unit group.
We introduce a combinatorial property for finitely generated groups called stackable that implies the existence of an inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. We also define algorithmically stackable groups, for which this procedure is an effective algorithm. This property gives a common model for algorithms arising from both rewriting systems and almost convexity for groups. We also introduce a new pair of asymptotic invariants that are filling inequalities refining the notions of intrinsic and extrinsic diameter inequalities fo…
Free-abelian by free groups: homomorphisms and algorithmic explorations
本文给出了自由阿贝尔乘自由群自同态的显式描述及其单射、满射判定，证明该类群是Hopfian且非coHopfian，并证明同构问题在有限作用下可判定、Brinkmann问题对单同态和自同构可判定。
Remarks and problems about algorithmic descriptions of groups
本文受Groves-Wilton定理启发，提出用带标记群同构类的编号格研究有限生成群的全局判定问题，建立Rice定理并给出有限可表示群的算法刻画，讨论Adian-Rabin定理的不足。
This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.
Complexity of some algorithmic problems in groups: a survey
本文综述了群论中字问题等算法问题在最坏、平均和一般情况下的复杂度，指出字问题平均复杂度通常为线性，而测地线问题等可低至次线性甚至常数。
This thesis contains a collection of algorithms for working with the twisted groups of Lie type known as Suzuki groups, and small and large Ree groups. The two main problems under consideration are constructive recognition and constructive membership testing. We also consider problems of generating and conjugating Sylow and maximal subgroups. The algorithms are motivated by, and form a part of, the Matrix Group Recognition Project. Obtaining both theoretically and practically efficient algorithms has been a central goal. The algorithms have been developed with, and implemented in, the computer…
Computing Joins in the Weak Order of Type B Coxeter Groups: an Algorithmic Approach
提出一种算法计算B型Coxeter群弱序中两个元素的并，将Markowsky算法从标准排列扩展到符号排列，并验证了Dyer关于这些并的几何解释的猜想。
The centrepiece of this paper is a normal form for primitive elements which facilitates the use of induction arguments to prove properties of primitive elements. The normal form arises from an elementary algorithm for constructing a primitive element p in F(x, y) with a given exponent sum pair (X, Y), if such an element p exists. Several results concerning the primitive elements of F(x, y) are recast as applications of the algorithm and the normal form.
We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if…
Semigroup algorithmic problems in metabelian groups
本文证明有限生成亚阿贝尔群的有限生成子半群上恒等元、群和逆元三个问题均可判定，并结合多学科工具建立对应关系，补全相关可判定性图景。
Recent advances in algorithmic problems for semigroups
本文综述了矩阵半群算法理论的最新进展，重点讨论低维矩阵群及具有交换性、幂零性和可解性等附加结构的群中各类算法问题的可判定性与复杂度。
Algorithmic aspects of left-orderings of solvable Baumslag--Solitar groups via its dynamical realization
本研究解答了Calderoni与Clay的问题，证明可解Baumslag-Solitar群$\text{BS}(1,n)$左序的共轭等价关系是超有限的，并依托其动力学实现的分类与能行性，研究该群左序的算法性质。
Motivated by a desire to test the security of the pubic key exchange protocol of I. Anshel, M. Anshel, and D. Goldfeld, (``An Algebraic Method for Public-Key Cryptography'', Mathematical Research Letters, vol. 6, pp. 1-5, 1999), we study algorithmic approaches to the simple commutator decision and promise problems (SCDP/SCPP) for the braid groups B_n. We take as our point of departure a seminal paper of O. Ore, (``Some Remarks on Commutators'', Proceedings of the American Mathematical Society, Vol. 2, No. 2, pp.307-314, 1951), which studies the SCPP for the symmetric groups. Our results build …
We consider decidability problems associated with Engel's identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) in groups generated by an automaton. We give a partial algorithm that decides, given $x,y$, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk's $2$-group is not Engel. We consider next the problem of recognizing Engel elements, namely elements $y$ such that the map $x\mapsto[x,y]$ attracts to $\{1\}$. Although this problem seems intractable in general, we prove that it is decidable for Grigorchuk's group: Engel eleme…
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…
An algorithmic approach for computing fundamental domains of crystallographic groups
提出一种基于Dirichlet胞腔的算法，通过有界长度的群元素推导半空间，计算晶体群的基本域，并应用于拓扑互锁结构。
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。
The central tree property and algorithmic problems on subgroups of free groups
该研究利用词元组的中心树性质，证明自由群子群均匀成员问题的平均情况复杂度远低于已知算法的最坏情况复杂度，并将结论推广至相对本原性问题，明确相关算法平均复杂度为仅依赖自由群秩的常数。
On isomorphisms to a free group and beyond
本文针对群论中难解的无限有限表现群同构问题，研究了有限表现群与自由群等特定群的同构判定，以及有限表现群嵌入给定极限群的算法问题。
We develop a general theory of "bisets": sets with two commuting group actions. They naturally encode topological correspondences. Just as van Kampen's theorem decomposes into a graph of groups the fundamental group of a space given with a cover, we prove analogously that the biset of a correspondence decomposes into a "graph of bisets": a graph with bisets at its vertices, given with some natural maps. The "fundamental biset" of the graph of bisets recovers the original biset. We apply these results to decompose the biset of a Thurston map (a branched self-covering of the sphere whose critica…
Algorithmic recognition of quasipositive braids of algebraic length two
本文提出判定辫子是否为两个标准生成元给定幂之共轭乘积的算法，并推广到一类 Garside 群，核心依据是循环滑动与左正规形式。
We study the interrelation of space functions of groups and the space complexity of the algorithmic word problem in groups.
The main result of this paper states that for any group $G$ with an automatic structure $L$ with unique representatives one can construct a uniform partial algorithm which detects $L$-rational subgroups and gives their preimages in $L$. This provides a practical, not just theoretical, procedure for solving the occurrence problem for such subgroups.
Following Craven and Rouquier's computational method to tackle Broué's abelian defect group conjecture, we present two algorithms implementing that procedure in the case of principal blocks of defect $D \cong C_{\ell} \times C_{\ell}$ for a prime $\ell$; the first describes a stable equivalence between $B_0(G)$ and $B_0(N_G(D))$, and the second tries to lift a such stable equivalence to a perverse derived equivalence. As an application, we show that Broué's conjecture holds for the principal $5$-block of the simple group $Ω^{+}_8(2)$.
Quadratic equations in metabelian Baumslag-Solitar groups
本文研究亚交换 Baumslag–Solitar 群 $\mathbf{BS}(1,n)$ 上二次方程的丢番图问题，证明 $n\neq\pm1$ 时其为 NP 完全，并刻画可定向、不可定向等子类的复杂性。
Algorithmic methods of finite discrete structures. Automorphism of Nonseparable Graphs
本文研究不可分图的自同构群构造问题，提出基于边割谱和边环谱的权重评估方法，通过轨道识别生成环，并给出示例验证方法有效性。
This is a survey of algorithmic problems in group theory, old and new, motivated by applications to cryptography.
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