On Dehn's algorithm

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46211/1/208_2005_Article_BF01361168.pd

On conjugacy separability of fibre products

In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group $G_1$ in which all finite index subgroups are conjugacy separable, but which has an index $2$ overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group $G_2$ which has a non-conjugacy separable subgroup of index $2$ such that every finite index normal overgroup of $G_2$ is conjugacy separable. The normality of the overgroup is essential in the last example, as such a group $G_2$ will always posses an index $3$ overgroup that is not conjugacy separable. Finally, we characterize $p$-conjugacy separable subdirect products of two free groups, where $p$ is a prime. We show that fibre products provide a natural correspondence between residually finite $p$-groups and $p$-conjugacy separable subdirect products of two non-abelian free groups. As a consequence, we deduce that the open question about the existence of an infinite finitely presented residually finite $p$-group is equivalent to the question about the existence of a finitely generated $p$-conjugacy separable full subdirect product of infinite index in the direct product of two free groups.Comment: v2: 38 pages; this is the version accepted for publicatio

Men Presenting With Sexual Thoughts of Children or Coercion: Flights of Fancy or Plans for Crime?

Introduction. There is limited evaluation of clinical and theoretical claims that sexual thoughts of children and coercing others facilitate sexual offending. The nature of these thoughts (what they contain) is also unknown. Aims. To examine the relationship between child/coercive sexual thoughts and sexual offending, and to determine the nature of these thoughts and any differences between sexual offending (SO), non-sexual offending (NSO) and non-offending (NO) men. Methods. In a cross-sectional computerized survey, anonymous qualitative and quantitative self-reported sexual thought and experience data were collected from 279 adult volunteers, comprising equal numbers of SO, NSO and NO men recruited from a medium-security UK prison and a community sample of 6081 men. Main Outcome Measures. Computerized Interview for Sexual Thoughts and Computerized Inventory of Sexual Experiences. Results. Three analytical approaches found child sexual thoughts were related to sexual offending; sexual thoughts with coercive themes were not. Latent class analyses identified three types of child sexual thought (primarily differentiated by interpersonal context: the reporting of own emotions, emotions of others or both) and four types of sexual thoughts of coercing others (chiefly discriminated by the other person’s response: no emotional states reported, consent, non-consent, mixed). Type of child sexual thought and participant group were not significantly related. Type of coercive sexual thought and group were marginally related; the consensual type was more common for the NO group, the non-consensual type more common for the SO group, than expected statistically. Conclusions. Child sexual thoughts are a risk factor for sexual offending and should be assessed by clinicians. Generally, sexual thoughts with coercive themes are not a risk factor, though thought type may be important (i.e. thoughts where the other person expresses an enduring lack of consent). Exploring the dynamic risk factors associated with each type of child/coercive thought may lead to more targeted treatment