J. -C. Birget, A. Yu. Olshanskii, E. Rips, M. Sapir
We prove that the word problem of a finitely generated group $G$ is in NP (solvable in polynomial time by a non-deterministic Turing machine) if and only if this group is a subgroup of a finitely presented group $H$ with polynomial isoperimetric function. The embedding can be chosen in such a way that $G$ has bounded distortion in $H$.
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On the Isoperimetric functions of a class of Artin Groups
本文针对定义图中不含标号为3的边的Artin群，利用相对扩展表示的small cancellation理论，构造出n^6多项式等周函数，从而证明偶Artin群的字问题可解。
Subgroups of word hyperbolic groups in dimension 2 over arbitrary rings
本文利用任意环上的等周不等式将Gersten关于2维字双曲群子群的双曲性结果推广至任意环，并给出了高阶双曲群的子群保持同阶双曲性的条件及组合定义。
We show that word-hyperbolic groups satisfy linear isoperimetric functions for all homotopy types of surface diagrams. This generalises the linear isoperimetric functions for disc and annular diagrams.
We prove that when n>=5, the Dehn function of SL(n,Z) is at most quartic. The proof involves decomposing a disc in SL(n,R)/SO(n) into a quadratic number of loops in generalized Siegel sets. By mapping these loops into SL(n,Z) and replacing large elementary matrices by "shortcuts," we obtain words of a particular form, and we use combinatorial techniques to fill these loops.
We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd-Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.
Conjugator Lengths and Isoperimetric functions
该研究构造了有限表示群，揭示共轭子长度函数与Dehn函数这两个不变量具有强独立性，还给出了共轭子长度函数递归但字、共轭问题均不可判定的首个例子。
Let $G$ be a finitely generated group equipped with a finite symmetric generating set and the associated word length function $|\cdot |$. We study the behavior of the probability of return for random walks driven by symmetric measures $μ$ that are such that $\sum ρ(|x|)μ(x)<\infty$ for increasing regularly varying or slowly varying functions $ρ$, for instance, $s\mapsto (1+s)^α$, $α\in (0,2]$, or $s\mapsto (1+\log (1+s))^ε$, $ε>0$. For this purpose we develop new relations between the isoperimetric profiles associated with different symmetric probability measures. These techniques allow us to …
We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize existence of primitives of the volume form with pre…
We consider finitely generated group endowed with a word metric. The group acts on itself by isometries, which induces an action on its horofunction boundary. The conjecture is that nilpotent groups act trivially on their reduced boundary. We will show this for the Heisenberg group. The main tool will be a discrete version of the isoperimetric inequality.
The computational complexity of the word problem in HNN-extension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory. It is shown that the word problem for an ascending HNN-extension of a group H is logspace reducible to the so-called compressed word problem for H. The main result of the paper states that the word problem for an HNN-extension of a hyperbolic group H with cyclic associated subgroups can be solved in polynomial time. This result can be easily extended to fundamental groups of graphs of groups with hyperbolic vertex groups and cyclic e…
This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every $m$ the first $m$ digits of a real number $α\ge 4$ are computable in time $\le C2^{2^{Cm}}$ for some constant $C>0$ then $n^α$ is equivalent (``big O'') to the Dehn function of a finitely presented group. The smallest isodiametric function of this group is $n^{3/4α}$. On the other hand if $n^α$ is equivalent to the Dehn function of a finitely presented group then the first $m$ digits of $α$ are computable in t…
We introduce the space function $s(n)$ of a finitely presented semigroup $S =<A\mid R>.$ To define $s(n)$ we consider pairs of words $w,w'$ over $A$ of length at most $n$ equal in $S$ and use relations from $R$ for the transformations $w=w_0\to...\to w_t= w'$; $s(n)$ bounds from above the tape space (or computer memory) sufficient to implement all such transitions $w\to...\to w'.$ One of the results obtained is the following criterion: A finitely generated semigroup $S$ has decidable word problem of polynomial space complexity if and only if $S$ is a subsemigroup of a finitely presented semigr…
The average-case complexity of the Word Problem for groups of matrices over~$\mathbb{Q}$ is linear
本文证明在位复杂度模型与输入字均匀随机选取的假设下，$\textsf{GL}_d(\mathbb{Z})$ 有限生成子群的字问题可在线性平均情况复杂度内解决，并推广至 $\textsf{GL}_d(R)$。
On the complexity of the word problem of the R. Thompson group V
本研究分析Lehnert与Schweitzer的证明，得出R. Thompson群V及其有限生成子群的字问题的复杂性，确定其属于对数DCFL且在确定性多带图灵机上具二次时间复杂度。
We establish the existence, finiteness, and uniqueness up to scaling of various isoperimetric profiles of a group, in all dimensions. We also show that these profiles all coincide in dimensions 4 and higher; in particular, the nth Dehn function is equal to FV^{n+1} for n at least 3. Even for dimension 3, there is significant overlap. When a group has decidable word problem, this has mild consequences for the growth rates of its profiles. We also establish a metric analogue for highly connected Riemannian manifolds.
In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup and, thus, an automaton semigroup with a PSPACE-complete word problem. We also show that there is an automaton group for which the word problem with a single rational constraint is PSPACE-complete. Additionally, we provide simpler constructions for …
We give a new characterization of primitive permutation groups tied to the notion of completely reachable automata. Also, we introduce sync-maximal permutation groups tied to the state complexity of the set of synchronizing words of certain associated automata and show that they are contained between the $2$-homogeneous and the primitive groups. Lastly, we define $k$-reachable groups in analogy with synchronizing groups and motivated by our characterization of primitive permutation groups. But the results show that a $k$-reachable permutation group of degree $n$ with $6 \le k \le n - 6$ is eit…
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…
Orbit-blocking words and the average-case complexity of Whitehead's problem in the free group of rank 2
该研究针对秩2自由群的Whitehead自同构问题，先证明轨道阻塞字的存在性，进而提出常数平均情况复杂度的判定算法。
Spectral Dehn functions and a characterisation of word-hyperbolicity
本文引入谱德恩函数，证明其与德恩函数的关系，并通过谱德恩函数的度量自由变体刻画词超几何性，同时展示其比德恩函数提供更细致的信息。
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。
We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a n…
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