We give a systematic exposition of memory-length algorithms for solving equations in noncommutative groups. This exposition clarifies some points untouched in earlier expositions. We then focus on the main ingredient in these attacks: Length functions.
After a self-contained introduction to Garside groups, we describe length functions induced by the greedy normal form and by the rational normal form in these groups, and compare their worst-case performances.
Our main concern is Artin's braid groups, with their two known Garside presentations, due to Artin and due to Birman-Ko-Lee (BKL). We show that in $B_3$ equipped with the BKL presentation, the (efficiently computable) rational normal form of each element is a geodesic, i.e., is a representative of minimal length for that element. (For Artin's presentation of $B_3$, Berger supplied in 1994 a method to obtain geodesic representatives in $B_3$.)
For an arbitrary number of strands, finding the geodesic length of an element is NP-hard, by a 1991 result of by Paterson and Razborov. We show that a good estimation of the geodesic length of a braid in Artin's presentation is measuring the length of its rational form in the \emph{BKL} presentation. This is proved theoretically for the worst case, and experimental evidence is provided for the generic case.
We show that the span of the variable $q$ in the Lawrence-Krammer-Bigelow representation matrix of a braid is equal to the twice of the dual Garside length of the braid, as was conjectured by Krammer. Our proof is close in spirit to Bigelow's geometric approach. The key observation is that the dual Garside length of a braid can be read off a certain labeling of its curve diagram.
We show that Lawrence's representation and linear representations from quantum sl_2 called generic highest weight vectors detect the dual Garside length of braids in a simple and natural way. That is, by expressing a representation as a matrix over a Laurent polynomial ring using certain natural basis, the span of the variable is equal to the constant multiples of the dual Garside length.
We present a solution to the conjugacy decision problem and the conjugacy search problem in Garside groups, which is theoretically simpler than the usual one, with no loss of efficiency. This is done by replacing the well known cycling and decycling operations by a new one, called cyclic sliding, which appears to be a more natural choice. We give an analysis of the complexity of our algorithm in terms of fundamental operations with simple elements, so our analysis is valid for every Garside group. This paper intends to be self-contained, not requiring any previous knowledge of prior algorithms…
In this paper, we obtain Groebner-Shirshov (non-commutative Gröbner) bases for the braid groups in the Birman-Ko-Lee generators enriched by new ``Garside word" $δ$. It gives a new algorithm for getting the Birman-Ko-Lee Normal Form in the braid groups, and thus a new algorithm for solving the word problem in these groups.
We present a simple construction which associates to every Garside group a metric space, called the additional length complex, on which the group acts. These spaces share important features with curve complexes: they are $δ$-hyperbolic, infinite, and typically locally infinite graphs. We conjecture that, apart from obvious counterexamples, additional length complexes are always of infinite diameter. We prove this conjecture for the classical example of braid groups $(B_n,Δ)$; moreover, in this framework, reducible and periodic braids act elliptically, and at least some pseudo-Anosov braids act…
Garside calculus is the common mechanism that underlies a certain type of normal form for the elements of a monoid, a group, or a category. Originating from Garside's approach to Artin's braid groups, it has been extended to more and more general contexts, the latest one being that of categories and what are called Garside families. One of the benefits of this theory is to lead to algorithms solving effectively the naturally occurring problems, typically the Word Problem. The aim of this paper is to present and solve these algorithmic questions in the new extended framework.
In this paper we show the statement in the title. To any Garside group of finite type, Wiest and the author associated a hyperbolic graph called the \emph{additional length graph} and they used it to show that central quotients of Artin-Tits groups of spherical type are acylindrically hyperbolic. In general, a euclidean Artin-Tits group is not \emph{a priori} a Garside group but McCammond and Sulway have shown that it embeds into an \emph{infinite-type} Garside group which they call a \emph{crystallographic Garside group}. We associate a \emph{hyperbolic} additional length graph to this crysta…
In this article, we introduce the notion of cycling operations of arbitrary order in Garside groups, which is a full generalization of the cycling and decycling operations. Theoretically, this notion together with other related concepts provides a context in which various definitions and arguments concerning Garside groups are unified and simplified as well as improved. Practically, it yields a new algorithm which has a considerably improved performance on solving the conjugacy problem of reducible braids.
Let $G$ be a Garside group endowed with the generating set $\mathcal{S}$ of non-trivial simple elements, and let $H$ be a parabolic subgroup of $G$. We determine a transversal $T$ of $H$ in $G$ such that each $θ\in T$ is of minimal length in its right-coset, $H θ$, for the word length with respect to $\mathcal{S}$. We show that there exists a regular language $L$ on $\mathcal{S} \cup \mathcal{S}^{-1}$ and a bijection $\mathrm{ev} : L \to T$ satisfying $\mathrm{lg} (U) = \mathrm{lg}_\mathcal{S}( \mathrm{ev}(U))$ for all $U \in L$. From this we deduce that the coset growth series of $H$ in $G$ i…
Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with $N$ strands and of Garside length $L$, the sliding circuit set should have at most $C\cdot L^{N-2}$ elements, for some constant $C$. We construct a family of braids which realise t…
We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conugacy problem given by the authors in a previous paper, are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.
The Garside group, as a generalization of braid groups and Artin groups of finite types, is defined as the group of fractions of a Garside monoid. We show that the semidirect product of Garside monoids is a Garside monoid. We use the semidirect product $\mathbb Z\ltimes G^n$ of the infinite cyclic group $\mathbb Z$ and the cartesian product $G^n$ of a Garside group $G$ to study the properties of roots and powers of elements in $G$. The main result is an estimate of the growth of the minimal word-length of powers of elements in Garside groups, when the generating set is the set of simple elemen…
We show that for each element $g$ of a Garside group, there exists a positive integer $m$ such that $g^m$ is conjugate to a periodically geodesic element $h$, an element with $|h^n|_\D=|n|\cdot|h|_\D$ for all integers $n$, where $|g|_\D$ denotes the shortest word length of $g$ with respect to the set $\D$ of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside group, its stable super summit set.
An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series…
We prove that the exponential growth rate of the regular language of penetration sequences is smaller than the growth rate of the regular language of normal form words, if the acceptor of the regular language of normal form words is strongly connected. Moreover, we show that the latter property is satisfied for all irreducible Artin monoids of spherical type, extending a result by Caruso. Our results establish that the expected value of the penetration distance $pd(x,y)$ in an irreducible Artin monoid of spherical type is bounded independently of the length of $x$, if $x$ is chosen uniformly a…
In this paper we study the reduction curves of a braid, and how they can be used to decompose the braid into simpler ones in a precise way, which does not correspond exactly to the decomposition given by Thurston theory. Then we study how a cyclic sliding (which is a particular kind of conjugation) affects the normal form of a braid with respect to the normal forms of its components. Finally, using the above methods, we provide the example of a family of braids whose sets of sliding circuits (hence ultra summit sets) have exponential size with respect to the number of strands and also with res…
We develop a theory of curve diagrams for Artin groups of type B. We define the winding number labeling and the wall crossing labeling of curve diagrams, and show that these labelings detect the classical and the dual Garside length, respectively. A remarkable point is that our argument does not require Garside theory machinery like normal forms, and is more geometric in nature.
In the general context of presentations of monoids, we study normalisation processes that are determined by their restriction to length-two words. Garside's greedy normal forms and quadratic convergent rewriting systems, in particular those associated with the plactic monoids, are typical examples. Having introduced a parameter, called the class and measuring the complexity of the normalisation of length-three words, we analyse the normalisation of longer words and describe a number of possible behaviours. We fully axiomatise normalisations of class (4, 3), show the convergence of the associat…
This paper is the second in a series in which the authors study the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in Garside groups. The ultra summit set USS(X) of an element X in a Garside group G is a finite set of elements in G, introduced by the second author, which is a complete invariant of the conjugacy class of X in G. A fundamental question, if one wishes to find bounds on the size of USS(X), is to understand its structure. In this paper we introduce two new operations on elements of USS(X), called `partial cycling' and `partial twisted decycling', and prove …
We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$…
Let $A\neq A_1, A_2, I_{2m}$ be an irreducible Artin--Tits group of spherical type. We show that periodic elements of $A$ and the elements preserving some parabolic subgroup of $A$ act elliptically on the additional length graph $\mathcal{C}_{AL}(A)$, an hyperbolic, infinite diameter graph associated to $A$ constructed by Calvez and Wiest to show that $A/Z(A)$ is acylindrically hyperbolic. We use these results to find an element $g\in A$ such that $\langle P,g \rangle\cong P* \langle g \rangle$ for every proper standard parabolic subgroup $P$ of $A$. The length of $g$ is uniformly bounded with…
Analysing statistical properties of the normal forms of random braids, we observe that, except for an initial and a final region whose lengths are uniformly bounded (that is, the bound is independent of the length of the braid), the distributions of the factors of the normal form of sufficiently long random braids depend neither on the position in the normal form nor on the lengths of the random braids. Moreover, when multiplying a braid on the right, the expected number of factors in its normal form that are modified, called the "expected penetration distance", is uniformly bounded. We explai…
Several key agreement protocols are based on the following "Generalized Conjugacy Search Problem": Find, given elements b_1,...,b_n and xb_1x^{-1},...,xb_nx^{-1} in a nonabelian group G, the conjugator x. In the case of subgroups of the braid group B_N, Hughes and Tannenbaum suggested a length-based approach to finding x. Since the introduction of this approach, its effectiveness and successfulness were debated. We introduce several effective realizations of this approach. In particular, a new length function is defined on B_N which possesses significantly better properties than the natural le…
For every finite Coxeter group $Γ$, each positive braids in the corresponding braid group admits a unique decomposition as a finite sequence of elements of $Γ$, the so-called Garside-normal form.The study of the associated adjacency matrix $Adj(Γ)$ allows to count the number of Garside-normal form of a given length.In this paper we prove that the characteristic polynomial of $Adj(B_n)$ divides the one of $Adj(B_{n+1})$. The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type $A$. We observe that this does not hold for type $D$. T…
It has been conjectured that in a braid group, or more generally in a Garside group, applying any sequence of monotone equivalences and word reversings can increase the length of a word by at most a linear factor depending on the group presentation only. We give a counter-example to this conjecture, but, on the other hand, we establish length upper bounds for the case when only right reversing is involved. We also state a new conjecture which would, like the above one, imply that the space complexity of the handle reduction algorithm is linear.
We define geodesic normal forms for the general series of complex reflection groups G(de,e,n). This requires the elaboration of a combinatorial technique in order to determine minimal word representatives and to compute the length of the elements of G(de,e,n) over some generating set. Using these geodesic normal forms, we construct intervals in G(e,e,n) that give rise to Garside groups. Some of these groups correspond to the complex braid group B(e,e,n). For the other Garside groups that appear, we study some of their properties and compute their second integral homology groups. Inspired by th…
We prove that, for any irreducible Artin-Tits group of spherical type $G$, the quotient of $G$ by its center is acylindrically hyperbolic. This is achieved by studying the additional length graph associated to the classical Garside structure on $G$, and constructing a specific element $x_G$ of $G/Z(G)$ whose action on the graph is loxodromic and WPD in the sense of Bestvina-Fujiwara; following Osin, this implies acylindrical hyperbolicity. Finally, we prove that "generic" elements of $G$ act loxodromically, where the word "generic" can be understood in either of the two common usages: as a res…
In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type $\tilde A_n$ we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin-Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter element in the Artin-Tits group. More precisely, if $W$ is a Coxeter group, one can consider the length $l_R$ on $W$ with respect to the generating set $R$ consisting of all reflections. Let $c$ be a Coxeter element in $W$ and let $P_c$ be the set of elements $p\in W$ such that …
The Levin conjecture was proposed by Levin in 1962 which conjectures the solvability of any group equation with coefficients in a torsion free group. The Levin conjecture is recently shown to hold for group equations of length seven by weight test and curvature distribution. These methods are applied on a group equation of length nine to explore the validity of Levin conjecture. It is found that the Levin conjecture has an affirmative answer for this equation modulo some exceptional cases.
The length-based approach is a heuristic for solving randomly generated equations in groups which possess a reasonably behaved length function. We describe several improvements of the previously suggested length-based algorithms, that make them applicable to Thompson's group with significant success rates. In particular, this shows that the Shpilrain-Ushakov public key cryptosystem based on Thompson's group is insecure, and suggests that no practical public key cryptosystem based on this group can be secure.
We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin--Tits monoid of spherical type (hence of each braid monoid) with respect to the standard generators, as the inverse of the determinant of a very simple matrix. Using this approach, we show that the exponential growth rates of the Artin--Tits monoids of type $A_n$ (positive braid monoids) tend to $3.233636\ldots$ as $n$ tends to infinity. This number is well-known, as …
Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words $u$ and $v$, up to taking cyclic permutations. This function might be bounded by a constant, even in the case when the standard conjugacy length function is unbounded. We give applications to the complexity of the conjugacy problem, estimating conjugacy growth rates, and languages. Our main result states that for a relatively hyperbolic group, the permutation conjugacy lengt…
We present an algorithm for solving the conjugacy search problem in the four strand braid group. The computational complexity is cubic with respect to the braid length.
