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Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden
... Firstly, it extends the support for string predicates from the SMT-LIB string theory standard [11] by (1) applying smarter and more specific axiom saturation and (2) adding support for their solving inside the decision procedure (e.g., for the ¬contains predicate). It also implements various optimizations (e.g., for regular constraints handling) and other decision procedures, e.g., the Nielsen transformation [32] for quadratic equations and a procedure for regular language (dis)equations; moreover, we added heuristics for choosing the best decision procedure to use. ...
... For an efficient handling of quadratic equations (systems of equations with at most two occurrences of each variable) with lengths, Noodler implements a decision procedure based on the Nielsen transformation [32]. The algorithm constructs a graph corresponding to the system and reasons about it to determine if the input formula is satisfiable or not [38,22]. ...
... Better results in AutomatArk and StringFuzz stem from the improvements in Mata and from heuristics tailored for regular expressions handling. Including Nielsen's algorithm [32] has the largest impact on the Kepler benchmark. The improvement on predicate-intensive benchmarks is caused by optimizations in axiom saturation for predicates. ...
Z3-Noodler is a fork of Z3 that replaces its string theory solver with a custom solver implementing the recently introduced stabilization-based algorithm for solving word equations with regular constraints. An extensive experimental evaluation shows that Z3-Noodler is a fully-fledged solver that can compete with state-of-the-art solvers, surpassing them by far on many benchmarks. Moreover, it is often complementary to other solvers, making it a suitable choice as a candidate to a solver portfolio.
... This rule performs a case split with respect to the possible alignment of the variables. The case-split rule is used in most, if not all, (semi-)decision procedures for string constraints, including Makanin's algorithm in (Makanin (1977)), Nielsen transformation (Nielsen (1917)) (also known as the Levi's lemma (Levi (1944))), and the procedures implemented in most state-ofthe-art solvers such as Z3 (Bjørner et al. (2009)), CVC4 (Liang et al. (2014)), Z3Str3 ), Norn (Abdulla et al. (2014)), and many more. In this paper, we will explain the general idea of our symbolic approach using the Nielsen transformation, which is the simplest of the approaches; nonetheless, we believe that the approach is applicable also to other procedures. ...
... Consider the word equation xz = yw, the primary type of atomic string constraints considered in this paper, where x, z, y, and w are word variables. When establishing satisfiability of the word equation, the Nielsen transformation (introduced in Nielsen (1917)) proceeds by first performing a case split based on the possible alignments of the variables x and y, the first symbol of the left and right-hand sides of the equation, respectively. More precisely, it reduces the satisfiability problem for xz = yw into satisfiability of (at least) one of the following four cases (1) y is a prefix of x, (2) x is a prefix of y, (3) x is an empty string, and (4) y is an empty string. ...
... In addition to our approach, it is also used as the basis of the work of Le and He (2018). On the other hand, the Nielsen transformation (Nielsen (1917)) is used by some tools that implement different approaches to discharge quadratic equations (e.g., Ostrich of Chen et al. (2019b)). Complex rewriting rules are used, e.g., when dealing with regular constraints in CVC5 ). ...
Case split is a core proof rule in current decision procedures for the theory of string constraints. Its use is the primary cause of the state space explosion in string constraint solving, since it is the only rule that creates branches in the proof tree. Moreover, explicit handling of the case split rule may cause recomputation of the same tasks in multiple branches of the proof tree. In this paper, we propose a symbolic algorithm that significantly reduces such a redundancy. In particular, we encode a string constraint as a regular language and proof rules as rational transducers. This allows us to perform similar steps in the proof tree only once, alleviating the state space explosion. We also extend the encoding to handle arbitrary Boolean combinations of string constraints, length constraints, and regular constraints. In our experimental results, we validate that our technique works in many practical cases where other state-of-the-art solvers fail to provide an answer; our Python prototype implementation solved over 50 % of string constraints that could not be solved by the other tools.
... straight-line of [9] or chain-free [9,10], which cover most of existing practical benchmarks), but even these are still PSPACE-complete (immediately due to Boolean combinations of regular constraints) and practically hard. Most of string solvers use base algorithms that resemble Makanin [8] or Nielsen's [11] algorithm in which word equations and regular constraints each generate one level of disjunctive branching, and the two levels multiply. Regular constraints particularly are considered complex and expensive, and reasoning with them is sometimes postponed and done only as the last step. ...
... For instance, in cases such as = ∧ ∈ * ∧ ∈ + + ∧ ∈ * , attempting to eliminate the equation results in an infinite case split (using, e.g., Nielsen's algorithm [11] or the algorithm of [12]) and it indeed leads to failure for all solvers we have tried. The regular constraints enforce UNSAT: since the on the left contains at least one , the on the right must answer with at least one ( has only 's). ...
... Our algorithm is not a complete alternative but a promising basis that could improve some of the existing solvers and become a core of a new one. With regard to equations and regular constraints, the fragment of chain-free constraints [10] that we handle, handled also by T , is the largest for which any string solvers offers formal completeness guarantees, with the exception of quadratic equations, handled, e.g., by [25,28], which are incomparable but of a smaller practical relevance (although some tools actually implement Nielsen's algorithm [11] to handle simple quadratic cases). The other solvers guarantee completeness on smaller fragments, notably that of OSTRICH (straight-line), N , and Z3 3RE; or use incomplete heuristics that work in practice (giving up guarantees of termination, over or under-approximating by various means). ...
When eating spaghetti, one should have the sauce and noodles mixed instead of eating them separately. We argue that also in string solving, word equations and regular constraints are better mixed together than approached separately as in most current string solvers. We propose a fast algorithm, complete for the fragment of chain-free constraints, in which word equations and regular constraints are tightly integrated and exchange information, efficiently pruning the cases generated by each other and limiting possible combinatorial explosion. The algorithm is based on a novel language-based characterisation of satisfiability of word equations with regular constraints. We experimentally show that our prototype implementation is competitive with the best string solvers and even superior in that it is the fastest on difficult examples and has the least number of timeouts.
... En témoigne l'article de 1941, écrit avec W. Adkisson, intitulé Extensions of homeomorphisms on the sphere. 20 Chapitre 1 : Les concepts de catégorie et de foncteur. ment d9un proessus rdu d9strtion des propriétés fondmentles qui permettent de dérire lirement l notion d9isomorphisme nturelF roessus qui ommené u printemps de IWRI qund iilenerg ssisté à un séminire de wvneD à l9niversité du wihignD sur l onstrution des extensions lgériques de ertins groupes 20 F in prtiulier wvne étudiit le suite exte ourte ...
... iilenerg mentionne à wvne que l9extension des groupes exposés iEdessus orE respond à l9homologie de teenrod pour le solénoïde pEdique 21 20 ...
... elon les délrtions fites pr gorry QPD il n9y ps de rtéristion unique de e qui est onsidéré omme une struture mthémtiqueF sl montre quelques visions informelles et formelles de e qu9on entend pr une struture mthémtique F hns ertins sD une struture mthémtique peut être vue omme un ensemle muni d9une série de reltions ssoiées 69 it dns les utres sD le onept de struture mthémtique n9est ps rédutile ux ensemles 70 F hns les deux sD une struture mthémtique peut être pensée omme une entitéD étudiée pr un domine des mthémtiquesD onstituée pr d9utres entités @de di'érent type 71 AD qui sont liés entre eux d9une ertine fçonF ge sont es liens qui déterminent une struture et qui permettent de l di'érenier d9une utre strutureF gonformément à e qui préèdeD di'érents uteurs soulignent l9importne des morE phismes d9une tégorie et des fonteurs entre des tégoriesD les deux vues omme des trnsformtions qui permettent l préservtion des propriétés struturellesF À e sujetD dns l9sntrodution de TD pF P il est mentionné que X he generl notion of tegory provides hrteriztion of the notion of strutureEpreserving trnsformtionD nd therey of speies of struE tures dmitting suh trnsformtionsF ewodey v plus loinD pre qu9il onsidère l9existene d9un dre référentiel su0E smment mple pour fonder plusieurs des strutures mthémtiques X le onept de 69. Voir, par exemple, la dénition d'espèce de structure Σ faite par Bourbaki [20,E IV .4]. Le concept de structure chez Boubaki ne joue pas un rôle décisif dans le développement des Éléments de Mathématique [32, p.330]. ...
L'objectif de cette thèse est de faire une étude historique et épistémologique de la notion d'adjonction dans le cadre de la théorie des catégories, afin de montrer son développement historique, sa relation avec d'autres concepts et méthodes mathématiques et, en fin de compte, de montrer son importance à la fois du point de vue mathématique et son intervention dans la discussion sur les fondements des mathématiques.L'analyse historique portera sur la période historique de 1945 à 1958, période qui va de la naissance des premiers concepts catégoriels par Samuel Eilenberg et Saunders MacLane à la publication de l'article Adjoint Functors par Daniel Kan.De même, on analysera certains contextes mathématiques associés au concept d'adjonction pour apprécier si dans de tels contextes le phénomène de l'adjonction était déjà présenté avant la publication de l'article de Kan
... We remark that determining whether an element is primitive or not is quite non-trivial, but decidable, as we shall see in §1.4.1. Nielsen, as part of a broader investigation, gave a necessary and sufficient condition for a pair of elements in F 2 to be a basis, using commutators: Proposition 1.2.4 (Dehn/Nielsen 9 , 1917 [Nie17]). Let w 1 , w 2 P F 2 " F pa, bq. ...
... The notation for the Christoffel words comes from the fact that they are all primitive elements of F pa, bq. Nielsen showed that the abelianisation map F Ñ Z 2 induces a bijection between conjugacy classes of primitive elements in F and primitive elements in Z 2 [Nie17]. Osborne-Zieschang showed in [OZ81] that the Christoffel words and their inverses provide a complete set of conjugacy class representatives for the primitive elements of F . ...
The theory of one-relator groups is now almost a century old. The authors therefore feel that a comprehensive survey of this fascinating subject is in order, and this document is an attempt at precisely such a survey. This article is divided into two chapters, reflecting the two different phases in the story of one-relator groups. The first chapter, written by the second author, covers the historical development of the theory roughly until the advent of geometric group theory. The second chapter, written by the first author, covers the recent progress in the theory up until the present day. The two chapters can be read independently of one another and have minimal overlap.
... On H * c (F (X, n)) it factors through the action of Out(F 2 ) via a homomorphism S 2 × S 3 → Out(F 2 ). Using Nielsen's identification Out(F 2 ) ∼ = GL 2 (Z) [Nie17], the latter homomorphism is the 2-dimensional representation sgn 2 ⊗ std 3 , where sgn 2 is the sign representation of the S 2 factor and std 3 = Z 3 /⟨(1, 1, 1)⟩ is the standard 2-dimensional representation of the S 3 factor. ...
... By [Nie17], abelianization gives Out(F 2 ) ∼ = GL 2 (Z). Since Theorem 1.5 shows that the associated graded gr H * c (F (X, n)) is a polynomial representation of GL 2 (Z), and since those are semi-simple, the collision filtration splits canonically so that H * c (F (X, n)) ∼ = gr H * c (F (X, n)) as S n × Out(F 2 )-representations. ...
... The notion of Nielsen equivalence goes back to the origins of geometric group theory and the work of Jakob Nielsen in the 1920s [19,20]. If G be a group, n ≥ 1 and T = (g 1 , . . . ...
... It is a consequence of Grushko-Neumann theorem [6,18] combined with Nielsen's results [19,20] that any generating pair of G is Nielsen equivalent to (s ν1 ...
We show that any non-standard generating pair of a hyperbolic triangle group is represented by a special almost orbifold covering with a good marking.
... Let F n be the free group of rank n and IA n be the subgroup of Aut(F n ) which induces the identity automorphism on Z n , the abelianization of F n . These groups have been studied by Nielsen in 1917 [19], but their structures remained not wellunderstood. In 1997, Krstić and McCool [16] proved that IA 3 is not finitely presentable and H 2 (IA 3 ) is not finitely generated. ...
... (1) When i = j ′ and j = i ′ , this case is essentially the same as χ 1,2 and χ 2,1 in IA 2 . Nielsen [19] proved that IA 2 is a free group generated by χ 1,2 and χ 2,1 and the result follows. (2) When i = i ′ and j = j ′ . ...
We investigate the action of Schur algebra on the Lie algebras of derivations of free Lie algebras and operad structures constructed from it. We also show that the Lie algebra of derivations is generated by quadratic derivations together with the action of the Schur operad. Applications to certain subgroups of the automorphism group of a finitely generated free group are given as well.
... The automorphism induced by the map that maps x 1 to x 2 and x 2 to x 1 is clearly not inner. Nielsen showed in [Nie18] that the free group F 2 is semicomplete. In the case where the rank of the free group is larger than two the group of IA-automorphisms is much larger than Inn(F n ). ...
... For a free group F 2 Nielsen showed in [Nie18] that IA(F 2 ) = Inn(F 2 ) and that the map Φ is also surjective, hence we have the following short exact sequence ...
In this short note we prove that a graph product of finitely generated abelian groups is semicomplete -- that is the kernel of the natural homomorphism induced by the abelianization of is equal to the inner automorphisms -- if and only if does not have a separating star.
... This states that for matrices A, B ∈ SL 2 (C) We also note that the action of Out(F 2 ) on Hom(F 2 , SL 2 (C))/ SL 2 (C) preserves the function θ → tr([θ(X), θ(Y )]). We rely here on an important fact about F 2 which has no counterpart for free groups of higher rank: given a basis X, Y for F 2 , every outer automorphism preserves the conjugacy class of XY X −1 Y −1 up to inversion [Nie17]. This, together with the fact that tr(A) = tr(A −1 ) for A ∈ SL 2 (C), implies that tr([θ(X), θ(Y )]) is an invariant function for Out(F 2 ). ...
... We now explain the relationship between Out(F 2 ) and GL 2 (Z). Any automorphism of F 2 preserves the commutator subgroup, and in particular Out(F 2 ) acts on the abelianization F ab 2 = F 2 /[F 2 , F 2 ] ∼ = Z 2 which is a free abelian group of rank 2. This action induces a map Out(F 2 ) → Aut(Z 2 ) = GL 2 (Z) and it is a theorem of Nielsen that this map is an isomorphism (see, for instance, [Aig13, Theorem 6.24] or [Nie17] for the original article). Thus GL 2 (Z) acts on the Markoff surface via the action of Out(F 2 ). ...
... The generating pair (t 1 , t 2 ) of π o 1 (O ) clearly gets mapped by η * = i * • η * onto the pair τ . Case (10). In this case we have p 1 = 2, p 2 = 3 and p 3 7 with (p 3 , 6) = 1, and T is equal to (s 1 s 2 s 1 s 2 2 , s 2 2 s 1 s 2 s 1 ). ...
... Case(10). η = i • η : F → S 2 defines a special almost orbifold covering of degree 6. ...
We extend the result of \cite{Dutra} to generating pairs of triangle groups, that is, we show that any generating pair of a triangle group is represented by a special almost orbifold covering.
... . . , f k ) generate the automorphism group Aut(F k ) of F k by the result of Nilesen [16], T-equivalent X , X ′ are Nielsen equivalent if φ = id G . Makoto Sakuma kindly pointed out to us that our generating pairs given in this paper are not pairwise T-equivalent. ...
We show that there are infinitely many Nielsen equivalence classes of the mapping class group of a closed oriented surface of genus at least eight.
... The chain-free fragment [9], which we extend in this paper, represents the largest fragment of string constraints for which any string solver offers formal completeness guarantees. Quadratic equations, addressed by tools like Retro [26,28] and Kepler 22 [49], are incomparable but have less practical relevance, though some tools, such as Z3-Noodler or OSTRICH, implement Nielsen's algorithm [63] to handle quadratic cases. Most other solvers guarantee completeness on smaller fragments (e.g., OSTRICH [56], Norn [7,8], and Z3str3RE [17]), or use incomplete heuristics that work in practice by over-/under-approximating or by sacrificing termination guarantees. ...
We introduce a novel decision procedure for solving the class of position string constraints, which includes string disequalities, not-prefixof, not-suffixof, str.at, and not-str.at. These constraints are generated frequently in almost any application of string constraint solving. Our procedure avoids expensive encoding of the constraints to word equations and, instead, reduces the problem to checking conflicts on positions satisfying an integer constraint obtained from the Parikh image of a polynomial-sized finite automaton with a special structure. By the reduction to counting, solving position constraints becomes NP-complete and for some cases even falls into PTime. This is much cheaper than the previously used techniques, which either used reductions generating word equations and length constraints (for which modern string solvers use exponential-space algorithms) or incomplete techniques. Our method is relevant especially for automata-based string solvers, which have recently achieved the best results in terms of practical efficiency, generality, and completeness guarantees. This work allows them to excel also on position constraints, which used to be their weakness. Besides the efficiency gains, we show that our framework may be extended to solve a large fragment of not-contains (in NExpTime), for which decidability has been long open, and gives a hope to solve the general problem. Our implementation of the technique within the Z3-Noodler solver significantly improves its performance on position constraints.
... for all n, c, with n ≥ 2. For c = 1 and n ≥ 2, we have rank(L 2 (IA(F n ))) = n 2 (n−1) 2 (see, [1,Theorem 5.1]). For n = 2 we have IA(F 2 ) = Inn(F 2 ), by a result of Nielsen [11] and by a result of Andreadakis [1, Theorem 6.1], we have rank(L c+1 (IA(F 2 ))) = 1 c d|c µ(d)2 c/d . For n = 3, by Theorem 1(2), we may give a lower bound of rank(L c+1 (IA(F 3 ))) in terms of the rank of L c 1 (H) for all c. ...
We prove that, for any positive integer c, the quotient group of the lower central series of the McCool group is isomorphic to two copies of the quotient group of the lower central series of a free group of rank 3 as -modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra of is naturally embedded into the Johnson Lie algebra of the IA-automorphisms of .
... According to a theorem of Nielsen [12], the commutator of two primitive elements is conjugate to [a, b] ±1 . ...
We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide explicit examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct quasimorphisms on and some of them are Calabi.
... It is clear that the inner automorphism group Inn F n of F n is contained in IA n . Nielsen [40] showed that IA 2 = Inn F 2 . For n ≥ 3, IA n is much larger than Inn F n . ...
In this paper we study the structure of the rational cohomology groups of the IA-automorphism group of a free group of rank three by using combinatorial group theory and representation theory. In particular, we detect non-trivial irreducible component in the second cohomology group of , which does not contained in the image of the cup product map of the first cohomology groups. We also show that the image of the triple cup product map of the first cohomology groups in the third cohomology group is trivial. As a corollary, we obtain that the fourth term of the lower central series of has finite index in that of the Andreadakis-Johnson filtration.
... The latter follows from a classic theorem of Nielsen [11]. ...
We study random nilpotent groups in the well-established style of random groups, by choosing relators uniformly among freely reduced words of (nearly) equal length and letting the length tend to infinity. Whereas random groups are quotients of a free group by such a random set of relators, random nilpotent groups are formed as corresponding quotients of a free nilpotent group. This model reveals new phenomena because nilpotent groups are not "visible" in the standard model of random groups (due to the sharp phase transition from infinite hyperbolic to trivial groups).
... It is an extension of the method proposed in [4] and implemented in Norn [5], referred to as the split algorithm. The split algorithm is, in turn, based on the well-known Nielsen transformation [40]. It builds a proof tree by iteratively applying a set of inference (split) rules on a word equation. ...
This paper proposes a Graph Neural Network-guided algorithm for solving word equations, based on the well-known Nielsen transformation for splitting equations. The algorithm iteratively rewrites the first terms of each side of an equation, giving rise to a tree-like search space. The choice of path at each split point of the tree significantly impacts solving time, motivating the use of Graph Neural Networks (GNNs) for efficient split decision-making. Split decisions are encoded as multi-classification tasks, and five graph representations of word equations are introduced to encode their structural information for GNNs. The algorithm is implemented as a solver named DragonLi. Experiments are conducted on artificial and real-world benchmarks. The algorithm performs particularly well on satisfiable problems. For single word \mbox{equations}, DragonLi can solve significantly more problems than well-established string solvers. For the conjunction of multiple word equations, DragonLi is competitive with state-of-the-art string solvers.
... The free group on A is written F (A), and we identify it with the set of reduced words. It is well known that every subgroup of F (A) is free [11]. It is also well-known that every finitely generated subgroup H of F (A) can be associated with a uniquely defined reduced rooted A-graph (Γ(H), 1), called the Stallings graph of H, with the following property: a reduced word is in H if and only if it labels a circuit in Γ(H) at vertex 1. ...
We study the average case complexity of the generalized membership problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the relative primitivity problem, using Shpilrain's recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group.
... The "tame automorphism problem" asks whether any automorphism is tame. Jung [103] and van der Kulk [203] proved this for n = 2, (also see [149,150] for free groups, [65] for free Lie algebras, and [67,141] for free associative algebras), so one takes n > 2. ...
This paper is the final part of a review of results concerning the quantization approach to the some classical aspects of noncommutative algebras. The first part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 213 (2022), pp. 110–144. The second part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 214 (2022), pp. 107–126. The third part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 215 (2022), pp. 95–128. The fourth part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 216 (2022), pp. 153–171.
... Note that this concept was studied by Shpilrain in [62], before being made explicit in [46,64]. It is well known due to Dehn and Nielsen that OEx 1 ; x 2 is a test word in F 2 (see [34,44]). Other examples of test words in F n were given by Zieschang [66,67], Rips [52], Dold [15], and Shpilrain [62]. ...
Let H be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system S of equations with constants from H is equivalent to a single equation. We also show that the algebraic set associated with S is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such an equation has the form w(x_1,\ldots,x_n)=h , where w\in F(X) , h\in H ).
From this we deduce the following statement: Let G be an arbitrary overgroup of the above group H . Then H is verbally closed in G if and only if it is algebraically closed in G .
These statements have interesting implications; here we give only two of them: If H is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from H is equivalent to a single equation. We give a positive solution to Problem 5.2 from the paper [J. Group Theory 17 (2014), 29–40] of Myasnikov and Roman’kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts.
Moreover, we describe solutions of the equation x^ny^m=a^nb^m in acylindrically hyperbolic groups (AH-groups), where a , b are non-commensurable jointly special loxodromic elements and n,m are integers with sufficiently large common divisor. We also prove the existence of special test words in AH-groups and give an application to endomorphisms of AH-groups.
... Remark 4.10. The condition that ϕ * (T) and T 0 are Nielsen equivalent implies that the morphism ϕ : B → A is π 1 -surjective, and that size(T 0 ) = size(T) rk(B). ...
We prove that any generating tuple of the fundamental group of a sufficiently large 2-dimensional orbifold is represented by an almost orbifold covering. As a corollary we obtain a generalization of Louder's Theorem which asserts that any two generating tuples of the fundamental group of a closed surface are Nielsen equivalent.
... Jung [99] and van der Kulk [196] proved this for n = 2, (also see [143,144] for free groups, [63] for free Lie algebras, and [65,135] for free associative algebras), so one takes n > 2. ...
The purpose of this review paper is the collection, systematization and discussion of recent results concerning the quantization approach to the Jacobian conjecture, as well as certain related topics.
... Äëÿ òåñòîâûõ ýëåìåíòîâ ñâîáîäíîé ãðóïïû èñïîëüçóþò òåðìèí òåñòîâîå ñëîâî. Ïåðâûé ïðèìåð òåñòîâîãî ñëîâà áûë ïîëó÷åí Íèëüñåíîì â 1918 ãîäó [111] Ïîçaeå áûëè óêàçàíû äðóãèå ïðèìåðû òåñòîâûõ ñëîâ [123,124,116,75,93,97]. ...
This monograph proposes the author's interpretation of the solvable
group theory which is one of the most important parts of the modern
group theory. It includes fundamentals as well as current achievements
the major part of which belongs to the author of the monograph.
The main objects of the investigation are the groups which are free in
varieties of solvable groups and their universal theories. Moreover, groups
of automorphisms of solvable groups and endomorphisms of solvable
groups are studied in the book.The monograph is intended for specialists
interested in algebra and the model theory.
... Äëÿ òåñòîâûõ ýëåìåíòîâ ñâîáîäíîé ãðóïïû èñïîëüçóþò òåðìèí òåñòîâîå ñëîâî. Ïåðâûé ïðèìåð òåñòîâîãî ñëîâà áûë ïîëó÷åí Íèëüñåíîì â 1918 ãîäó [111] Ïîçaeå áûëè óêàçàíû äðóãèå ïðèìåðû òåñòîâûõ ñëîâ [123,124,116,75,93,97]. ...
This monograph proposes the author's interpretation of the solvable
group theory which is one of the most important parts of the modern
group theory. It includes fundamentals as well as current achievements
the major part of which belongs to the author of the monograph.
The main objects of the investigation are the groups which are free in
varieties of solvable groups and their universal theories. Moreover, groups
of automorphisms of solvable groups and endomorphisms of solvable
groups are studied in the book.The monograph is intended for specialists
interested in algebra and the model theory.
... Let F n be the free group of rank n with basis x 1 ; : : : ; x n . Nielsen [28] showed that IA.F 2 / coincides with the inner automorphism group of F 2 . In 1935, Magnus [22] showed that IA.F n / is finitely generated. ...
In this paper, we show that the third subgroup of the Andreadakis–Johnson filtration of the automorphism group of a free group coincides with the third group of the lower central series of the IA-automorphism group.
Z3-Noodler is a fork of the Z3 SMT solver replacing its string theory implementation with a portfolio of decision procedures and a selection mechanism for choosing among them based on the features of the input formula. In this paper, we give an overview of the used decision procedures, including a novel length-based procedure, and their integration into a robust solver with a good overall performance, as witnessed by Z3-Noodler winning the string division of SMT-COMP’24 by a large margin. We also extended the solver with a support for model generation, which is essential for the use of the solver in practice.
We study the average case complexity of the Uniform Membership Problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the Relative Primitivity Problem, using Shpilrain’s recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group.
We present a new algorithm for solving string constraints. The algorithm builds upon a recent method for solving word equations and regular constraints that interprets string variables as languages rather than strings and, consequently, mitigates the combinatorial explosion that plagues other approaches. We extend the approach to handle linear integer arithmetic length constraints by combination with a known principle of equation alignment and splitting, and by extension to other common types of string constraints, yielding a fully-fledged string solver. The ability of the framework to handle unrestricted disequalities even extends one of the largest decidable classes of string constraints, the chain-free fragment. We integrate our algorithm into a DPLL-based SMT solver. The performance of our implementation is competitive and even significantly better than state-of-the-art string solvers on several established benchmarks obtained from applications in verification of string programs.
This paper is the fourth part of a review of results concerning the quantization approach to the some classical aspects of noncommutative algebras. The first part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 213 (2022), pp. 110–144. The second part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 214 (2022), pp. 107–126. The third part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 215 (2022), pp. 95–128. The final part of the survey will be published in the next issue.
Keywords: automorphism, quantization, Jacobian conjecture
We argue that in string solving, word equations and regular constraints are better mixed together than approached separately as in most current string solvers. We propose a fast algorithm, complete for the fragment of chain-free constraints, in which word equations and regular constraints are tightly integrated and exchange information, efficiently pruning the cases generated by each other and limiting possible combinatorial explosion. The algorithm is based on a novel language-based characterisation of satisfiability of word equations with regular constraints. We experimentally show that our prototype implementation is competitive with the best string solvers and even superior in that it is the fastest on difficult examples and has the least number of timeouts.
A homogeneous quasimorphism on a normal subgroup N of G is said to be G-invariant if for every and for every . Invariant quasimorphisms have naturally appeared in symplectic geometry and the extension problem of quasimorphisms. Moreover, it is known that the existence of non-extendable invariant quasimorphisms is closely related to the behavior of the stable mixed commutator length , which is a certain generalization of the stable commutator length . In this survey, we review the history and recent developments of invariant quasimorphisms and stable mixed commutator length. The topics we treat include several examples of invariant quasimorphisms, Bavard's duality theorem for invariant quasimorphisms, Aut-invariant quasimorphisms, and the estimation of the dimension of spaces of non-extendable quasimorphisms. We also mention the extension problem of partial quasimorphisms.
The purpose of this review is the collection and systematization of results concerning the quantization approach to the some classical aspects of non-commutative algebras, especially to the Jacobian conjecture. We start with quantization proof of Bergman centralizing theorem, then discourse authomorphisms of INd-schemes authomorphisms, then go to aproximation issues. Last chapter dedicated to relations between -theory Burnside type theorems and Jacobian Conjecture (Jagzev approach). This issue contains the second part of the work. The first part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 213 (2022), pp. 110-144. Continuation will be published in future issues.
The purpose of this review is the collection and systematization of results concerning
the quantization approach to the some classical aspects of non-commutative algebras, especially to
the Jacobian conjecture. We start with quantization proof of Bergman centralizing theorem, then
discourse authomorphisms of INd-schemes authomorphisms, then go to aproximation issues. Last
chapter dedicated to relations between P I-theory Burnside type theorems and Jacobian Conjecture
(Jagzev approach). This issue contains the first part of the work; continuation will be published in
future issues.
Keywords and phrases: automorphism, quantization, Jacobian conjecture.
AMS Subject Classification: 14R10, 18G85
In this short note we prove that a graph product GΓ of finitely generated abelian groups is semicomplete – that is the kernel of the natural homomorphism Aut(GΓ)→Aut(GΓab) induced by the abelianization of GΓ is equal to the inner automorphisms – if and only if Γ does not have a separating star.
Given a finitely generated group Γ, a directed graph Λ, and a map R:Λ→Γ, we introduce the notion of an (R,Λ)-directed Anosov representation. This is a weakening of the notion of Anosov representations. Our main theorem gives a procedure to construct (R,Λ)-directed Anosov representations using Fock-Goncharov positivity. As an application of our main theorem, we construct large families of primitive stable representations from F2 to PGL(V), including non-discrete and non-faithful examples.
Le groupe de tresses soudées pures sur n-brins ou groupe de McCool PΣn possède des familles de sous-groupes telles que le groupe de tresses pures ordinaires Pn, le groupe de McCool triangulaire supérieur PΣ^+_{n} et le groupe des automorphismes intérieurs partiels In. Il est établi que la filtration d’Andreadakis restreinte à PΣ^+_n et celle restreinte à Pn coïncident chacune avec sa série centrale descendante. Nous démontrons un résultat analogue pour In et nous établissons également un isomorphisme entre In et Int(PΣ^+_{n+1}), le groupe des automorphismes intérieurs de PΣ^+_{n+1}.Nous construisons ensuite une variante de la représentation de Burau du groupe de tresses virtuelles sur n-brins VBn qui ne se factorise pas en une représentation du groupe de tresses soudées WBn. Enfin, dans le cas du groupe de tresses soudées pures ou groupe de McCool PΣn, nous proposons une procédure itérative comparable à celle de la méthode de Long-Moody qui permet de construire une représentation matricielle fidèle de PΣn généralisant ainsi une extension de la représentation de Gassner de Pn.
In the present paper, we construct a variant of the Burau representation of two generalizations of the classical braid group. For the Gassner representation, we propose an iterative procedure to find and generalize the extension of this representation.
Let be a free group of finite rank . We prove that if H is a subgroup of with and R is a retract of , then is a retract of H. However, for every and every , there exist a subgroup H of of rank m and a retract R of of rank k such that is not a retract of H. This gives a complete answer to a question of Bergman. Furthermore, we prove that for every family S of endomorphisms of and every subgroup H of with .
Case split is a core proof rule in current decision procedures for the theory of string constraints. Its use is the primary cause of the state space explosion in string constraint solving, since it is the only rule that creates branches in the proof tree. Moreover, explicit handling of the case split rule may cause recomputation of the same tasks in multiple branches of the proof tree. In this paper, we propose a symbolic algorithm that significantly reduces such a redundancy. In particular, we encode a string constraint as a regular language and proof rules as rational transducers. This allows to perform similar steps in the proof tree only once, alleviating the state space explosion. In our preliminary experimental results, we validated that our technique (implemented in a Python prototype) works in many practical cases where other state-of-the-art solvers, such as CVC4 or Z3, fail to provide an answer.
We consider a certain subgroup An+ of the automorphism group of a free group of rank n. It can be regarded as a free group analogue of the group Λn of integral lower-triangular matrices. We call An+ the lower-triangular automorphism group of a free group. The first aim of the paper is to give a finite presentation for An+.
The abelianization of the free group induces the surjective homomorphism ρ+ from An+ to Λn. In our previous paper [18], we introduced the lower-triangular IA-automorphism group IAn+. Here we show that IAn+ coincides with the kernel of ρ+. The second aim of the paper is to give an infinite presentation for IAn+.
Finally, we study a relation of the second homology groups between An+ and Λn. In particular, we compute the second homology group H2(Λn,L) by using Magnus's presentation where L is a principal ideal domain in which 2 is invertible. For example, L=Q,Z/pZ for prime p≥3. This gives a lower bound on the integral second homology group of An+.
For the action of the outer automorphism group of the rank two free group on the corresponding variety of P S L ( 2 , C ) \mathsf {PSL}(2,\mathbb {C}) characters, two domains of discontinuity have been known to exist that are strictly larger than the set of Schottky characters. One was introduced by Bowditch in 1998 (followed by Tan, Wong, and Zhang in 2008) and the other by Minsky in 2013. We prove that these two domains are equal. We then show that they are contained in the set of characters having what we call the bounded intersection property.
Groups St Andrews 2017 in Birmingham - edited by C. M. Campbell April 2019
Let be a free group of finite rank . We prove that if H is a subgroup of with and R is a retract of , then is a retract of H. However, for every and every , there exist a subgroup H of of rank m and a retract R of of rank k such that is not a retract of H. This gives a complete answer to a question of Bergman. Furthermore, we provide positive evidence for the inertia conjecture of Dicks and Ventura. More precisely, we prove that for every family S of endomorphisms of and every subgroup H of with .
meine Schrift „Kurvenetze auf Flächen
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