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Free group

In abstract algebra, a free group on a set X is the group F(X) generated by the elements of X such that every element can be uniquely represented as a reduced word in the generators and their inverses, with no additional relations beyond the group axioms.^[1] Elements of F(X) are equivalence classes of finite sequences from X \cup X^{-1}, where equivalence is defined by freely inserting or deleting pairs xx^{-1} for x \in X \cup X^{-1}, and the group operation is concatenation followed by reduction to eliminate such pairs.^[2] The empty word serves as the identity element, and the inverse of a word s_1 \cdots s_n is s_n^{-1} \cdots s_1^{-1}.^[2] Free groups embody the "freest" possible non-abelian group structure on a given generating set, making them fundamental in combinatorial group theory and algebraic topology.^[3] The rank of a free group is the cardinality of its free basis X, an invariant under isomorphism: two free groups are isomorphic if and only if they have bases of the same cardinality.^[1] For example, the free group on one generator is isomorphic to the integers \mathbb{Z} under addition, while on two or more generators, it is non-abelian and infinite.^[2] A key universal property characterizes free groups: for any group H and any function \phi: X \to H, there exists a unique group homomorphism \phi^*: F(X) \to H extending \phi.^[1] Notable theorems highlight their role in group theory. The Nielsen-Schreier theorem states that every subgroup of a free group is itself free, with rank satisfying a specific formula relating the ranks of the supergroup and subgroup.^[3] Additionally, every group is a homomorphic image of some free group, underscoring free groups as universal building blocks for constructing other groups via quotients by normal subgroups.^[3] Free groups appear in applications such as knot theory, where the fundamental group of the complement of a knot in three-space is often free or related to free groups.^[3]

Definition and Basics

Definition

In group theory, the free group on a set S provides an intuitive model of the most unrestricted group structure generated by S, imposing no relations among the generators beyond the fundamental axioms of groups, such as associativity, identity, and inverses. This construction captures the essence of generation without additional constraints, allowing elements to be formed freely through products of generators and their inverses, subject only to the necessary cancellations inherent in group multiplication.^[1]^[4] Formally, given a set S, let S^{-1} denote a disjoint set of formal symbols representing the inverses of elements in S, and let A = S \cup S^{-1}. The free group F(S) is constructed as the quotient of the free monoid on the alphabet A (consisting of all finite words, including the empty word as identity) by the congruence relation generated by free cancellations, where adjacent pairs of the form s s^{-1} or s^{-1} s (for s \in S) are identified with the empty word. Equivalently, the elements of F(S) can be represented uniquely as reduced words over A, which are nonempty finite sequences from A containing no such cancellable adjacent pairs, with the group operation defined by concatenation followed by reduction to eliminate any new cancellable pairs. The identity is the empty word, and the inverse of a reduced word is obtained by reversing it and replacing each symbol with its inverse counterpart.^[1]^[4]^[5] The set S serves as a free generating set, or basis, for F(S), meaning every element of F(S) can be uniquely expressed as a reduced word in the elements of S and their inverses, with no further relations holding among them. For a finite set S with |S| = n, the free group is denoted F_n, emphasizing its rank n, the cardinality of the basis. This definition presupposes familiarity with basic group concepts, including generators and the role of relations in presentations.^[1]^[4]

Examples

The free group on zero generators, denoted F_0, is the trivial group consisting solely of the identity element.^[6] The free group on one generator, F_1, is isomorphic to the additive group of integers \mathbb{Z}, where the generator corresponds to the element 1 in \mathbb{Z}.^[6] The free group on two generators, F_2, is generated by elements a and b, with group elements represented as reduced words in a, a^{-1}, b, b^{-1}; for instance, the word aba^{-1}b^{-1} is a typical non-identity element.^[6] Unlike abelian groups, F_2 is non-abelian, as demonstrated by the relation ab \neq ba.^[6] In topology, the fundamental group of the wedge of two circles (also known as the figure-eight space) is isomorphic to F_2, reflecting the two independent loops based at the wedge point.^[7] The free group on countably infinite generators, denoted F_\infty, is generated by a countable set \{x_1, x_2, x_3, \dots \}, with elements formed by finite reduced words over these generators and their inverses, yielding an uncountably infinite group despite the countable basis.^[1]

Historical Development

Origins in the early 20th century

The development of the concept of free groups in the early 20th century was closely tied to the maturation of abstract group theory, particularly through combinatorial approaches to understanding group presentations and relations. Building on Walther von Dyck's late-19th-century work on Fuchsian groups and the idea of groups defined solely by generators without additional relations, mathematicians in the 1900s sought to formalize these "relation-free" structures as foundational objects in group theory. Dyck's 1882 and 1883 papers emphasized such groups as having the simplest presentations, providing an early motivation for exploring them in the context of discontinuous transformation groups.^[8]^[9] This emergence was driven by the growing field of combinatorial group theory, which aimed to analyze groups via their generating sets and the absence of imposed relations, allowing any group to be viewed as a quotient of a free group by a normal subgroup generated by relations. Early motivations arose from topological problems, such as determining the fundamental groups of surfaces and solving word problems, where the lack of relations enabled systematic manipulation of group elements as reduced words. Max Dehn's 1911 work on the word problem for surface groups exemplified this approach, using combinatorial techniques to study groups without extraneous relations.^[10]^[8]^[11] The terminology and notation for these structures solidified in the 1920s, with Otto Schreier introducing and employing the term "freie Gruppe" (free group) in his 1927 paper on subgroups of free groups, where he provided an algebraic proof of key results. Jakob Nielsen's earlier 1921 paper had already laid groundwork by proving that finitely generated subgroups of free groups are free, though Schreier's work formalized the notation in a broader algebraic context.^[12]^[13]^[14] Partial inspirations for free groups also stemmed from studies of permutation groups and symmetric groups, which served as models for abstracting infinite discrete actions without fixed relations, influencing the combinatorial viewpoint in early group theory. Kurt Reidemeister further integrated free groups into knot theory in his 1932 book, treating them comprehensively as tools for analyzing presentations. Nielsen's contributions, such as his work on free products, briefly underscored their role in decomposing more complex groups.^[10]^[15]

Key contributors and milestones

Jakob Nielsen made foundational contributions to the theory of free groups during the 1910s and 1920s. In his 1921 paper published in Matematisk Tidsskrift, he introduced Nielsen transformations, a set of elementary operations on generating sets of subgroups of free groups that preserve the subgroup generated. These transformations enable the conversion of any finite generating set into a canonical form, facilitating the proof that finitely generated subgroups of free groups are free and establishing a normal form for elements in free groups.^[14] Building on Nielsen's work, Otto Schreier advanced the understanding of free group subgroups in the mid-1920s. In his 1927 paper "Die Untergruppen der freien Gruppe," based on his 1926 habilitation thesis, Schreier proved that every subgroup of a free group is itself free, extending Nielsen's result to infinitely generated subgroups. This theorem, known as the Nielsen-Schreier theorem, provided a combinatorial framework for describing subgroup structure via the Schreier transversal method and had profound implications for the algebraic properties of free groups.^[13] The 1930s saw significant milestones in the study of free products, closely related to free groups. Aleksandr Kurosh developed the theory of free products in papers published in 1932 and 1934, culminating in the Kurosh subgroup theorem, which states that every subgroup of a free product of free groups is itself a free product of free groups and a conjugate of a subgroup of one of the factors. This result generalized aspects of free group structure to amalgamated constructions and influenced subsequent work in combinatorial group theory.^[16] Free groups also gained prominence in algebraic topology during the 1930s through connections to fundamental groups. Herbert Seifert, in his 1930 thesis and 1931 publication, and Egbert van Kampen, in his 1933 paper "On the Connection between the Fundamental Groups of Some Related Spaces," established the Seifert-van Kampen theorem, which computes the fundamental group of a space as a free product with amalgamation of the fundamental groups of its subspaces. This theorem highlighted free groups as models for fundamental groups of wedges of circles, bridging abstract algebra and topology. Post-World War II developments emphasized computational aspects of free groups within the emerging field of computational group theory. In the 1950s and 1960s, algorithms like the Todd-Coxeter procedure, partially implemented by B. Haselgrove in 1953 on the EDSAC computer, enabled coset enumeration for subgroups of free groups, allowing practical verification of freeness and rank computations. The 1967 Oxford conference spurred further advances, including the Knuth-Bendix completion algorithm in 1967 for string rewriting in free groups, facilitating automated normal form calculations and decision procedures. These tools laid the groundwork for modern computer algebra systems like GAP, initiated in 1986.^[17] Later, around 1945, Alfred Tarski posed influential problems concerning the elementary theory of non-abelian free groups, sparking decades of research on decidability and logical properties in group theory.^[18]

Constructions

Via reduced words

One standard combinatorial construction of the free group on a set S of generators begins by forming an alphabet consisting of the elements of S together with their formal inverses S^{-1} = \{s^{-1} \mid s \in S\}, where S \cap S^{-1} = \emptyset.^[19] Reduced words are then defined as the finite sequences (including the empty sequence) from S \cup S^{-1} that contain no adjacent pairs of the form ss^{-1} or s^{-1}s for any s \in S.^[19] The free group F(S) is the set of all such reduced words, equipped with a group operation that makes it into a group with S as a free generating set.^[19] The group multiplication is defined by concatenating two reduced words and then applying free reduction, which repeatedly cancels any adjacent inverse pairs until no such pairs remain.^[19] Formally, if u and v are reduced words, their product uv is the reduced word obtained from the concatenation u \cdot v by this cancellation process.^[20] The identity element is the empty word, and the inverse of a reduced word w = x_1 x_2 \cdots x_n (with each x_i \in S \cup S^{-1}) is the reduced word w^{-1} = x_n^{-1} \cdots x_2^{-1} x_1^{-1}, where s^{-1} and (s^{-1})^{-1} = s for s \in S.^[19] This operation is associative, as verified by the standard properties of word concatenation and reduction.^[19] A key feature of this construction is the normal form theorem, which states that every element of F(S) corresponds uniquely to a reduced word: distinct reduced words represent distinct group elements.^[19] This uniqueness ensures that the construction yields a well-defined group without relations among the generators other than those imposed by the inverses.^[20] This syntactic approach via reduced words provides a concrete realization of the free group, complementary to more abstract perspectives such as the universal property.^[19]

Universal property

The free group F(S) on a set S is characterized by the following universal mapping property: for any group G and any function f: S \to G, there exists a unique group homomorphism \phi: F(S) \to G such that \phi(s) = f(s) for all s \in S.^[21] To see this, one realizes F(S) explicitly as the group of reduced words over S \cup S^{-1}, where the homomorphism \phi is constructed by sending each generator s \in S to f(s) and extending multiplicatively to words, with well-definedness ensured by the reduction process that eliminates inverses; uniqueness follows since any homomorphism is determined by its values on the generators S, with no further relations imposed.^[21] This property implies that F(S) is initial in the category of groups equipped with a distinguished generating set isomorphic to S, and more categorically, the free group functor is left adjoint to the forgetful functor from the category of groups to sets, yielding a natural isomorphism

\Hom_{\Grp}(F(S), G) \cong \Hom_{\Set}(S, G)

for any group G, where the bijection associates to each set map its unique group homomorphism extension.^[22]^[21] In the category of groups, F(S) also realizes the coproduct of |S| copies of the infinite cyclic group \mathbb{Z}, underscoring its role as the universal object freely generated by S.^[21]

Properties

Fundamental properties

Free groups exhibit several core algebraic properties that stem directly from their construction as groups freely generated by a basis. A free group F on a basis X has the property that for any subset Y \subseteq X, the subgroup generated by Y is free on exactly that subset Y, as elements of this subgroup consist solely of reduced words using letters from Y and their inverses, without relations involving elements outside Y^[23]. This reflects the absence of relations beyond inverses in the free generation. The rank of a free group is defined as the cardinality of any basis; all bases for a given free group have the same cardinality, ensuring uniqueness up to isomorphism for free groups of a fixed rank n^[23]. For a free group of finite rank n, the generating set consists of n basis elements, but considering inverses, the associated symmetric alphabet has $2n symbols used in word representations. Free groups of rank n \geq 2 are non-abelian, as the commutator [a, b] = aba^{-1}b^{-1} of two distinct basis elements a, b is nontrivial and cannot reduce to the identity under the free group relations^[23]. Moreover, the center Z(F_n) of a free group F_n of rank n \geq 2 is trivial, meaning Z(F_n) = \{e\}, since any central element would commute with all basis elements, forcing it to be the identity in the reduced word presentation^[24]. Free groups are residually finite, meaning for any nontrivial element g \in F, there exists a finite quotient group F/N (with N a normal subgroup) such that the image of g is nontrivial, allowing finite quotients to separate distinct elements^[25]. This property arises from the topological realization of free groups as fundamental groups of graphs, enabling constructions of finite-index subgroups via coverings.

Key theorems

The Nielsen-Schreier theorem asserts that every subgroup of a free group is itself free.^[26] This result, fundamental to the structure of free groups, implies that the subgroup lattice of a free group consists entirely of free groups, preserving the absence of nontrivial relations within subgroups. A brief proof sketch relies on the topological realization of free groups as fundamental groups of graphs: the free group F corresponds to the fundamental group of a wedge of circles (a graph \Gamma), and a subgroup H \leq F is the fundamental group of the covering space \tilde{\Gamma} \to \Gamma corresponding to H, which is also a graph whose fundamental group is free.^[26] When the index [F : H] is finite, say equal to n, and F has rank r, the theorem provides an explicit rank formula for H: \operatorname{rank}(H) = nr - n + 1.^[26] This formula arises combinatorially from the covering space: \tilde{\Gamma} has n vertices and nr edges, a spanning tree uses n-1 edges, and the remaining nr - n + 1 edges generate H freely. The significance lies in quantifying the "complexity" of subgroups, enabling computations of ranks in quotients and embeddings, which underpin classifications in combinatorial group theory.^[26] The Kurosh subgroup theorem extends this to free products, stating that if H is a subgroup of the free product \bigstar_{i \in I} A_i of nontrivial groups A_i, then H is isomorphic to a free product F \bigstar (\bigstar_{k \in K} H_k), where F is a free group and each H_k is a conjugate of a nontrivial subgroup of some A_i.^[27] Here, the free product \bigstar_{i \in I} A_i is the "freest" amalgamated combination of the A_i with no relations beyond those within each factor. A topological proof interprets the free product as the fundamental group of a wedge of spaces with fundamental groups A_i, and H as the fundamental group of a covering space, decomposing into free and conjugated components via Bass-Serre theory. This theorem classifies subgroups of free products, generalizing the Nielsen-Schreier result and facilitating the study of splittings in group actions on trees.^[27] The ping-pong lemma provides a criterion for freeness via group actions: if a group G generated by a set S acts on a set X such that for each s \in S \cup S^{-1}, there are nonempty disjoint subsets X_s \subset X satisfying s \cdot X_t \subset X_s for all t \in S \cup S^{-1} \setminus \{s^{-1}\} and a basepoint o \in X \setminus \bigcup X_s with s \cdot o \in X_s for all s \in S \cup S^{-1}, then G is freely generated by S.^[28] In the context of actions on trees (where free actions imply freeness of stabilizers), this lemma detects free subgroups by ensuring generators "play ping-pong" without fixed points or overlaps, preventing relations. Its significance is in embedding free groups into larger structures, such as linear groups like \mathrm{SL}(2, \mathbb{Z}), supporting Tits' alternative that finitely generated linear groups are either virtually solvable or contain free subgroups of rank at least two.^[28] These theorems collectively enable the classification of subgroups in free groups and their generalizations, underpinning embeddings and structural decompositions in geometric and combinatorial group theory.

Relations to Other Groups

Free abelian groups

A free abelian group on a set S, denoted \mathbb{Z}^{(S)}, is constructed as the direct sum \bigoplus_{s \in S} \mathbb{Z}, consisting of all formal integer linear combinations \sum_{s \in S} k_s s where k_s \in \mathbb{Z} and only finitely many k_s are nonzero, with addition defined componentwise.^[29] The elements of S serve as a free abelian basis, meaning every element can be uniquely expressed as such a finite sum, and the group operation is additivity rather than the concatenation used in non-abelian free groups.^[29] This structure arises as the abelianization of the free group F(S) generated by S, obtained by quotienting F(S) by its commutator subgroup [F(S), F(S)], the normal subgroup generated by all commutators [g, h] = g h g^{-1} h^{-1} for g, h \in F(S).^[30] The resulting quotient F(S)/[F(S), F(S)] is isomorphic to \mathbb{Z}^{(S)}, with the explicit isomorphism mapping each reduced word in F(S) to the vector of exponent sums for each generator in S.^[30] For instance, if S = \{x_1, \dots, x_n\} and a reduced word is w = x_1^{e_1} x_2^{e_2} \cdots x_n^{e_n} (up to reordering, which is irrelevant in the quotient), it maps to (e_1, e_2, \dots, e_n) \in \mathbb{Z}^n.^[30] Free abelian groups are torsion-free, containing no nontrivial elements of finite order, as they embed into direct products of copies of \mathbb{Z}, which itself has no torsion.^[31] Unlike general abelian groups, they also admit divisible quotients; for example, when S is countably infinite, \mathbb{Z}^{(S)} surjects onto \mathbb{Q} via a map sending basis elements to rational multiples that generate the additive structure of \mathbb{Q}.^[32] This property highlights their role as projective objects in the category of abelian groups, facilitating resolutions and extensions in homological algebra.^[33]

Free products

The free product of two groups G and H, denoted G * H, is the group generated by the elements of G and H with no relations imposed between elements of G and H other than those already present within each group itself. More precisely, it is the coproduct in the category of groups, consisting of all finite alternating products (reduced words) of non-identity elements from G and H, such as g_1 h_1 g_2 h_2 \cdots g_n h_n where g_i \in G \setminus \{e\}, h_i \in H \setminus \{e\}, and the sequence begins with either a g or an h. The group operation is concatenation followed by reduction, which combines consecutive elements from the same factor group using its own multiplication and cancels identities. This construction ensures that G and H embed injectively into G * H via canonical inclusions.^[19] The universal property of the free product characterizes it up to isomorphism: for any group K and homomorphisms \phi: G \to K, \psi: H \to K, there exists a unique homomorphism \theta: G * H \to K such that \theta \circ i_G = \phi and \theta \circ i_H = \psi, where i_G and i_H are the inclusions. In equation form, this is expressed as \operatorname{Hom}(G * H, K) \cong \{\,(\phi, \psi) \mid \phi: G \to K, \psi: H \to K\,\}. This property extends to free products of arbitrary families of groups. For free groups specifically, if F(S) and F(T) are free groups on disjoint sets S and T, then F(S) * F(T) \cong F(S \cup T), as both are isomorphic to the free product of |S \cup T| copies of the infinite cyclic group \mathbb{Z}. Thus, free products of free groups are themselves free.^[19]^[34] Elements of a free product admit unique normal forms via syllable decomposition: every non-identity element can be uniquely written as an alternating product of nontrivial syllables, where each syllable is a nontrivial element from one factor group, and no two consecutive syllables come from the same factor. This normal form theorem facilitates computations and proofs of properties, such as the embeddability of the factor groups and the fact that subgroups of free products satisfy specific structural theorems, including those describing their decomposition into free factors and conjugates of subgroups of the original factors.^[19]^[34]

Advanced Topics

Tarski's problems

In the mid-1940s, Alfred Tarski posed several foundational questions about the first-order logic of non-abelian free groups, including whether distinct non-abelian free groups of different ranks are elementarily equivalent—meaning they satisfy exactly the same first-order sentences—and whether the elementary theory of a non-abelian free group is decidable, i.e., whether there exists an algorithm to determine the truth of any first-order sentence in the language of groups. These problems concerned the shared first-order properties of free groups.^[18] The key challenges centered on the decidability of the elementary theory, which encompasses quantifiers over group elements and operations, building on earlier decidability results for existential fragments like the word problem in free groups. In contrast to the undecidable elementary theory of groups in general, partial progress toward Tarski's problems included decidability of the universal theory of free groups by Makanin in 1982, but the full elementary theory remained open.^[35] These problems were resolved affirmatively in the 2000s: Olga Kharlampovich and Alexei Myasnikov announced initial results in 1998 and provided a full proof in 2006 showing that all non-abelian free groups are elementarily equivalent and that their common elementary theory is decidable. Independently, Zlil Sela established the same results in 2006 using methods from hyperbolic geometry and Diophantine sets over groups, confirming no other infinite groups share this exact elementary theory except certain limit groups.^[35]^[36]^[37] Despite some technical critiques of the 2006 Kharlampovich-Myasnikov proof by Sela, the core theorems hold, marking a major milestone around 2006 rather than later dates.^[38] It is worth distinguishing these logical problems from "Tarski monster groups," another class named after Tarski, which are infinite non-abelian groups—distinct from free groups—where every proper nontrivial subgroup is cyclic of a fixed prime order; their existence was conjectured by Tarski in the 1940s and constructed by Alexander Olshanskii in the 1980s.^[39] The resolution of Tarski's free group problems imposes limits on algorithmic group theory by revealing the boundaries of decidability in logical fragments, while enabling computational verification of first-order properties in free groups that underpin applications in geometric group theory.^[18]

Applications in topology and geometry

Free groups play a central role in algebraic topology as fundamental groups of specific spaces. The fundamental group of the wedge sum (or bouquet) of n circles, denoted \bigvee_{i=1}^n S^1, is isomorphic to the free group F_n on n generators.^[19] This isomorphism arises from the Seifert-van Kampen theorem, which computes the fundamental group of a space obtained by gluing two path-connected open sets along a path-connected subspace; applying it iteratively to the wedge sum yields the free group structure, where each circle contributes a generator corresponding to a loop around it.^[19] The theorem's amalgamation over the trivial intersection at the basepoint ensures no relations are imposed beyond those from the individual circles.^[19] In geometric group theory, free groups are characterized by their actions on trees. A finitely generated free group F_n acts freely and properly discontinuously on its Cayley graph with respect to a free basis, which is a tree with vertices corresponding to group elements and edges labeled by generators. This action is simplicial, meaning stabilizers of vertices are trivial, and the quotient graph is a single edge per generator. Bass-Serre theory provides a framework for understanding such actions: a group acts freely on a tree if and only if it is free, linking the combinatorial structure of the tree to the group's presentation. These tree actions encode splittings of free groups and extend to broader classes via graphs of groups, where edge stabilizers correspond to subgroups amalgamated in free products. Free groups also embed naturally into the geometry of non-positively curved spaces. Trees, as the Cayley graphs of free groups, are CAT(0) spaces, satisfying the CAT(0) inequality for geodesic triangles, which implies unique geodesics and a well-defined convex hull.^[40] More generally, free groups act geometrically (properly and cocompactly) on CAT(0) cube complexes of dimension 1, namely their Cayley trees viewed as cubical complexes, where hyperplanes are midpoints of edges separating the tree into components.^[41] This cubical perspective highlights free groups as special cases of right-angled Artin groups, facilitating studies of hyperplane stabilizers and combinatorial curvature.^[41] In modern geometric group theory, free groups exemplify relatively hyperbolic groups. A free group F_n is hyperbolic (hence relatively hyperbolic relative to the empty collection of subgroups) due to its \delta-hyperbolic Cayley tree, where geodesic triangles are \delta-thin for some \delta > 0.^[42] Relative hyperbolicity extends this to settings where free groups serve as peripheral subgroups in cusped structures, such as in fundamental groups of manifolds with toroidal cusps; post-2000 developments, including Bowditch's combinatorial formulation, emphasize electrifications and coned-off Cayley graphs to capture relative quasiconvexity.^[42] These tools address algorithmic and rigidity questions in higher-rank settings, contrasting with purely hyperbolic behavior.^[42] Links to undecidability appear in geometric contexts, where free group properties via Tarski problems inform questions about 3-manifold groups acting on hyperbolic spaces.^[42]

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The abelianization of the free group on n generators is Zn. Redefinition. G is generated by n elements if there is a surjection: (∗) q : Fn → G from the ...<|control11|><|separator|>
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[PDF] free product factorization - UChicago Math
Aug 20, 2008 · The free product of two groups G1 and G2, denoted G1 ∗ G2 is defined as the unique group H such that there exist injective homomorphisms i1 :.
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tarski's problem about the elementary theory of free groups has a ...
Dec 14, 1998 · Moreover, the elementary theory of a finitely generated free semigroup of rank at least two is also undecidable [29]. During the last 50 years ...Missing: undecidability | Show results with:undecidability
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Elementary theory of free non-abelian groups - ScienceDirect.com
Aug 15, 2006 · Tarski's problem about the elementary theory of free groups has a positive solution. Electron. Res. Announc. Amer. Math. Soc., 4 (December ...Missing: undecidability | Show results with:undecidability<|control11|><|separator|>
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Diophantine geometry over groups VI: the elementary theory of a ...
Jun 29, 2006 · This paper is the sixth in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such ...
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[PDF] A report on TARSKI'S DECIDABILITY Problem: ”ELEMENTARY ...
This paper contains a list of crucial mistakes and counterexamples to some of the main statements in the paper ”Elementary theory of free nonabelian groups” by ...
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Tarski group - PlanetMath
Mar 22, 2013 · A Tarski group is an infinite group G such that every non-trivial proper subgroup. of G is of prime order. Tarski groups are also called Tarski ...
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Metric Spaces of Non-Positive Curvature - SpringerLink
Free delivery 14-day returnsThe purpose of this book is to describe the global properties of complete simply connected spaces that are non-positively curved in the sense of AD Alexandrov.
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[PDF] CAT(0) Cube Complexes and Groups - Math
On the other hand, CAT(0) cube complexes have a combinatorial nature that give them the look and feel of trees. The added structure provided by hyperplanes has ...
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RELATIVELY HYPERBOLIC GROUPS B. Farb 1 Introduction
In this paper we introduce a theory of relatively hyperbolic groups. Fundamental groups of complex (resp. quaternionic, Cayley) hyperbolic manifolds with ...Missing: URL | Show results with:URL