Abstract: The opportunity approach to compliance focuses on understanding how rule-breaking behaviour takes place and then tries to reduce the factors that enable rule breaking. This chapter reviews two core criminological theories within the opportunity approach: routine activity theory and situational crime prevention. The chapter assesses empirical evidence as to whether policies based on these theories can reduce rule-violating behaviour. Moreover, it discusses the extent to which the opportunity approach can result in displacement and adaptation effects. And, finally, it explores potential downsides to the opportunity approach such as victim blaming and reductions in autonomy and freedom of choice.

Introduction

Alfred Tarski in 1940 made three well-known conjectures concerning nonabelian free groups (see Section 2). There had been various partial solutions until complete positive solutions were presented during the past 15 years by Kharlampovich and Myasnikov (see [51]–[59]) and independently by Z. Sela (see [78]–[83]). In the Kharlampovich- Myasnikov approach the proof arose from a detailed study of fully residually free groups (called limit groups in Sela's approach), the development of algebraic geometry over free groups, and an elimination process involving solutions of equations over free groups based on work of Makhanin and Razborov (see [51]–[59]). These steps were mirrored, with somewhat different terminology, by Sela, who called his approach diophantine geometry over free groups.

The positive solution of the Tarski conjectures provides a straightforward proof of Magnus's theorem in surface groups which we present. This result was proved directly by J. Howie [46] and independently by O. Bogopolski [10]. We will present this proof in Section 4. This type of proof leads to several different types of questions.

1. • Which additional nontrivial free group results are true in surface groups but difficult to obtain directly?

2. • What first-order properties of nonabelian free groups are true beyond the class of elementary free groups?

After showing a proof of Magnus's Theorem based on the solution of the Tarski problems we give several examples of other free group results holding in surface groups. Using this technique we give a proof of a theorem of D. Lee on C-test words. We then consider and prove certain other results that hold in elementary free groups, in particular surface groups, including the retract theorem of Turner [86] and the property of conjugacy separability.

After this we turn to the second type of question and survey a large number of recent results. In particular we first consider groups satisfying certain quadratic properties that we call Lyndon properties and show that the class of groups satisfying these properties are closed under many amalgam constructions.

Abstract

Discriminating groups were introduced by Baumslag, Myasnikov and Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups was the main method of attack used by Kharlampovich and Myasnikov in their solution of the celebrated Tarski conjectures. The class of discriminating groups, however, has taken on a life of its own and has been an object of a considerable amount of study. In this paper we survey the large array of results concerning the class of discriminating groups that have been developed over the past decade.

Introduction

Discriminating groups were introduced by Baumslag, Myasnikov and Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups was the main method of attack used by O. Kharlampovich and A. Myasnikov in their solution of the celebrated Tarski conjectures. The class of discriminating groups, however, has taken on a life of its own and has been an object of a considerable amount of study. In this paper we survey the large array of results concerning the class of discriminating groups that have been developed over the past decade.

In Section 1, we define discrimination for groups and describe its ties to other areas. Also the concept of trivially discriminating (TD) groups is introduced, and the concept of squarelike groups is defined. It is also indicated how to define discrimination for arbitrary algebraic systems. It is also shown how to generalize the concept of squarelike to arbitrary algebras.

Abstract

Remarkable ties between group theory, logic and algebraic geometry have come to light via the positive solution of the Tarski conjecture. This has led to further work on the universal theory of groups. In this paper we describe and survey this material a large body of which is not familiar to most group theorists

Contents

1. Introduction.

2. First Order Languages and Model Theory.

3. The Tarksi Problems.

4. Residually Free and Universally Free Groups.

5. Algebraic Geometry over Groups and Applications.

6. The Positive Solution to the Tarski Problems.

7. Discriminating, Co-discriminating and Squarelike Groups.

8. Open Questions.

Introduction

The elementary theory of a group G consists of all the first-order or elementary sentences (see section 2) which are true in G. Although this is a concept which originated in formal logic, in particular model theory, it arises independently from the theory of equations within groups. Recall that an equation in a group G is a word W(x1,…, xn, g1,…, gk) in free variables x1,…, xn and constants g1,…, gk which are elements of G. A solution consists of an n-tuple (h1,…, hn) of elements from G which upon substitution for x1,…, xn make the word trivial in G. Hence an equation is a first-order sentence in the language L[G] consisting of the elementary language of group theory (again see section 2) augmented by allowing constants from the group G.

Introduction

Group theoretical investigation and classification efforts take advantage of the various group theoretical constructions: the attempt is to recapture information about groups of interest from their various subgroups, homomorphic images and extensions. The particularly successful theory of one-relator groups - which greatly benefitted from the pioneering work of W. Magnus - provided motivation for investigating one-relator quotients of free products as well as extensions and automorphisms of free groups. Questions of SQ-universality and recognizability, stemming from decision problems, also added interest in one-relator groups possessing free quotients, and in various groups with free subgroups.

Recall that if P is a group property, then a group G is virtuallyP if it has a subgroup of finite index satisfying P. Alternatively we also call G a virtual P-group. G is P-by-finite if G has a normal subgroup of finite index satisfying P. If P is a subgroup inherited property, such as torsion-freeness, freeness, or solvability then virtually P and P-by-finite are equivalent.

The structure of virtually-free groups {free-by-finite groups} is rather well understood {see Section 3} and generalizes the structure of free groups in expected ways. Virtually free groups also have connections with automatic groups and hyperbolic groups {see Section 3}.

The present paper represents the start of a general program to extend knowledge about virtually one-relator groups. The aim is to line up known results and give some extensions of relevance to the proposed project. Since a good number of properties of free groups appear in one-relator groups it is hoped that a similar parallelism can be detected for virtually one-relator groups.