# The Elementary Theory of Groups

## Benjamin Fine, Anthony Gaglione, Alexei Myasnikov, Gerhard Rosenberger, Dennis Spellman

- 321 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

# The Elementary Theory of Groups

## Benjamin Fine, Anthony Gaglione, Alexei Myasnikov, Gerhard Rosenberger, Dennis Spellman

## About This Book

After being an open question for sixty years the Tarski conjecture was answered in the affirmative by Olga Kharlampovich and Alexei Myasnikov and independently by Zlil Sela. Both proofs involve long and complicated applications of algebraic geometry over free groups as well as an extension of methods to solve equations in free groups originally developed by Razborov. This book is an examination of the material on the general elementary theory of groups that is necessary to begin to understand the proofs. This material includes a complete exposition of the theory of fully residually free groups or limit groups as well a complete description of the algebraic geometry of free groups. Also included are introductory material on combinatorial and geometric group theory and first-order logic. There is then a short outline of the proof of the Tarski conjectures in the manner of Kharlampovich and Myasnikov.

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## Information

# 1 Group theory and logic: introduction

# 1.1 Group theory and logic

_{v}} for G from which any element of G can be generated as a word or expression in the {g

_{v}} together with a set of relations on these generators from which any part of the group table can be constructed. In Chapter 2 we will examine combinatorial group theory in detail. Although a group presentation is a succinct way to express a group, it was clear from the beginnings of the discipline, that working with group presentations required some detailed algorithmic knowledge and certain decision questions.

_{v}) is an arbitrary word in the generators of G, can one decide algorithmically, in a finite number of steps, whether W(g

_{v}) represents the identity in G or not. If such an algorithm exists we say that G has a solvable word problem. If not, G has an unsolvable word problem. Dehn presented a geometric method to show that the fundamental group of an orientable surface of genus g â„ 2, which we denote by S

_{g}, has a solvable word problem. In particular he gave an algorithm which systematically reduced the length of any word equal to the identity in Ï(S

_{g}). If a particular wordâs length is greater than 1 and cannot be reduced then that word does not represent the identity. Such an algorithm is now called a Dehn algorithm. Subsequently small cancellation theory, (see Chapter 3), was developed to determine additional groups that have Dehn algorithms. More recently it was shown that finitely presented groups with Dehn algorithms are precisely the word-hyperbolic groups of Gromov (see Chapter 2). In 1955 Novikov

^{[205]}and independently Boone

^{[34]}

^{[35]}proved that, in general. the word problem is unsolvable, that is, there exist finitely presented groups with unsolvable word problems. Hence questions about word problems now focus on which particular classes of groups have solvable word problems. As decribed by Magnus (see

^{[178]}), given the Novikov-Boone result, any solution of the word problem is actually a triumph over nature.