Zlil Sela is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution [1] of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first-order theories of finitely generated non-abelian free groups. [2]
Sela received his Ph.D. in 1991 from the Hebrew University of Jerusalem, where his doctoral advisor was Eliyahu Rips. Prior to his current appointment at the Hebrew University, he held an Associate Professor position at Columbia University in New York. [3] While at Columbia, Sela won the Sloan Fellowship from the Sloan Foundation. [3] [4]
Sela gave an Invited Address at the 2002 International Congress of Mathematicians in Beijing. [2] [5] He gave a plenary talk at the 2002 annual meeting of the Association for Symbolic Logic, [6] and he delivered an AMS Invited Address at the October 2003 meeting of the American Mathematical Society [7] and the 2005 Tarski Lectures at the University of California at Berkeley. [8] He was also awarded the 2003 Erdős Prize from the Israel Mathematical Union. [9] Sela also received the 2008 Carol Karp Prize from the Association for Symbolic Logic for his work on the Tarski conjecture and on discovering and developing new connections between model theory and geometric group theory. [10] [11]
Sela's early important work was his solution [1] in mid-1990s of the isomorphism problem for torsion-free word-hyperbolic groups. The machinery of group actions on real trees, developed by Eliyahu Rips, played a key role in Sela's approach. The solution of the isomorphism problem also relied on the notion of canonical representatives for elements of hyperbolic groups, introduced by Rips and Sela in a joint 1995 paper. [12] The machinery of the canonical representatives allowed Rips and Sela to prove [12] algorithmic solvability of finite systems of equations in torsion-free hyperbolic groups, by reducing the problem to solving equations in free groups, where the Makanin–Razborov algorithm can be applied. The technique of canonical representatives was later generalized by Dahmani [13] to the case of relatively hyperbolic groups and played a key role in the solution of the isomorphism problem for toral relatively hyperbolic groups. [14]
In his work on the isomorphism problem Sela also introduced and developed the notion of a JSJ-decomposition for word-hyperbolic groups, [15] motivated by the notion of a JSJ decomposition for 3-manifolds. A JSJ-decomposition is a representation of a word-hyperbolic group as the fundamental group of a graph of groups which encodes in a canonical way all possible splittings over infinite cyclic subgroups. The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groups [16] and this work gave rise a systematic development of the JSJ-decomposition theory with many further extensions and generalizations by other mathematicians. [17] [18] [19] [20] Sela applied a combination of his JSJ-decomposition and real tree techniques to prove that torsion-free word-hyperbolic groups are Hopfian. [21] This result and Sela's approach were later generalized by others to finitely generated subgroups of hyperbolic groups [22] and to the setting of relatively hyperbolic groups.
Sela's most important work came in early 2000s when he produced a solution to a famous Tarski conjecture. Namely, in a long series of papers, [23] [24] [25] [26] [27] [28] [29] he proved that any two non-abelian finitely generated free groups have the same first-order theory. Sela's work relied on applying his earlier JSJ-decomposition and real tree techniques as well as developing new ideas and machinery of "algebraic geometry" over free groups.
Sela pushed this work further to study first-order theory of arbitrary torsion-free word-hyperbolic groups and to characterize all groups that are elementarily equivalent to (that is, have the same first-order theory as) a given torsion-free word-hyperbolic group. In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word-hyperbolic group then G is word-hyperbolic as well.
Sela also proved that the first-order theory of a finitely generated free group is stable in the model-theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory.
An alternative solution for the Tarski conjecture has been presented by Olga Kharlampovich and Alexei Myasnikov. [30] [31] [32] [33]
The work of Sela on first-order theory of free and word-hyperbolic groups substantially influenced the development of geometric group theory, in particular by stimulating the development and the study of the notion of limit groups and of relatively hyperbolic groups. [34]
Theorem. Two non-abelian torsion-free hyperbolic groups are elementarily equivalent if and only if their cores are isomorphic. [35]
In mathematics, the free groupFS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms. The members of S are called generators of FS, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses.
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
Eliyahu Rips is an Israeli mathematician of Latvian origin known for his research in geometric group theory. He became known to the general public following his co-authoring a paper on what is popularly known as Bible code, the supposed coded messaging in the Hebrew text of the Torah.
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act.
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology, and combinatorial group theory. In a very influential chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.
In model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley rank.
Brian Hayward Bowditch is a British mathematician known for his contributions to geometry and topology, particularly in the areas of geometric group theory and low-dimensional topology. He is also known for solving the angel problem. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick.
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated applications of the operations of free product with amalgamation and HNN extension, via the notion of the fundamental group of a graph of groups. Bass–Serre theory can be regarded as one-dimensional version of the orbifold theory.
In the mathematical area of geometric group theory, a Van Kampen diagram is a planar diagram used to represent the fact that a particular word in the generators of a group given by a group presentation represents the identity element in that group.
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group G has more than one end if and only if G admits a nontrivial action on a simplicial tree with finite edge-stabilizers and without edge-inversions.
James W. Cannon is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.
In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other. Small cancellation conditions imply algebraic, geometric and algorithmic properties of the group. Finitely presented groups satisfying sufficiently strong small cancellation conditions are word hyperbolic and have word problem solvable by Dehn's algorithm. Small cancellation methods are also used for constructing Tarski monsters, and for solutions of Burnside's problem.
In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group in terms of the length of that relation. The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive. The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function that estimates the area of a minimal surface in a Riemannian manifold in terms of the length of the boundary curve of that surface.
Mladen Bestvina is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.
In the mathematical subject of geometric group theory, the Baumslag–Gersten group, also known as the Baumslag group, is a particular one-relator group exhibiting some remarkable properties regarding its finite quotient groups, its Dehn function and the complexity of its word problem.
Olga Kharlampovich is a Russian-Canadian mathematician working in the area of group theory. She is the Mary P. Dolciani Professor of Mathematics at the CUNY Graduate Center and Hunter College.
In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991.
In the mathematical subject of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf.
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