Mind Your Expectations: Exploring the Roles of Suggestion and Intention in Mindfulness Training

Mindfulness training (MT) has received increasing recognition for its therapeutic benefits in a variety of clinical contexts. Despite acknowledgement that MT effects are predicated upon the development of both mindful attention and intention, research on MT mechanisms has focused chiefly upon attentional effects. By contrast, hypnosis research has focused explicitly on suggestion techniques for cultivating beneficial therapeutic expectations. Comparing similarities between mindfulness and hypnosis techniques, this paper explores mechanisms of suggestion tacitly employed in mindfulness interventions. Distinctions between mindfulness meditation and hypnotic induction are then used to identify a form of intentionality that is unique to MT, including candidate markers of mindful intention that may help to explain mindfulness’ salutary effects. Finally, the idea of changing intentions during MT is discussed, generating suggestions for how best to monitor the interaction between expectation and attentional practice when studying mindfulness interventions. Studies of intention and expectation in MT could help to determine: i) the degree to which MT benefits are driven by expectation effects rather than changes to attention, ii) how to best motivate the development of mindful attention in therapeutic interventions, and iii) what factors predict the generalization of mindfulness techniques to improve emotion regulation. By acknowledging that suggestion may be important for cultivating mindful intentions, it may be possible to deepen our understanding of how to optimally deliver mindfulness training and improve participant well-being

On abstract commensurators of groups

We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated. This applies in particular to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2. We also construct a finitely generated, torsion-free group which can be mapped onto Z and which has a finitely generated commensurator.Comment: 13 pages, no figur

Topological Designs

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is large, both bounds are the best known for large genus. In genus one and two, we solve the problem exactly. Our methods generalize to variants in which the allowed number of pairwise intersections is odd, even, or bounded, and to surfaces with boundary components.Comment: 14 p., 4 Figures. To appear in Geometriae Dedicat

A Combination Theorem for Metric Bundles

We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles (including exact sequences of groups) that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.Comment: v3: Major revision: 56 pages 5 figures. Many details added. Characterization of convex cocompact subgroups of mapping class groups of surfaces with punctures in terms of relative hyperbolicity given v4: Final version incorporating referee comments: 63 pages 5 figures. To appear in Geom. Funct. Ana

Personalising Practice Using Preferences for Meditation Anchor Modality

Many people are starting to establish contemplative practices and Mindfulness-Based Interventions have become quite popular. While Mindfulness-Based Interventions positively impact well-being, drop-out and lack of practice-maintenance plagues these interventions. Such adherence issues may reveal a lack of fit between participant partiality for attentional anchors of meditative practice and the intervention’s use of the breath as the anchor of attention. No study had yet compared partiality towards practices using anchors from different sensory modalities (e.g., auditory and visual) thus the present study examined such individual differences, sharing resources on the Open Science Framework1. Participants (N = 82) engaged 10-min practices within three modalities (somatosensory, auditory, and visual) and partiality towards these meditations was modelled. Partiality differences did exist: 49% preferred the breath, 30% the auditory-phrase, and 21% the visual-image. Pre-practice motivation and anchor-modality predicted partiality while cardiac responses were also positively associated with partiality. Preferences were updated through experience and over half of participants left the experiment partial to a different anchor than their initial meditation-naïve bias. Tangible next-steps are discussed, including integrating additional anchor modalities into existing interventions by offering brief practices with a variety of anchors. Suggestions are made for increasing post-training contact using email-automation to answer central practice-maintenance questions, including whether and which contemplative benefits are predicated on continued practice

Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator

A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.Comment: 11 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V3: revised following referee's comment

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = -1

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t = −1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added

Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in (relatively) hyperbolic groups". We study there the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. In the second part, called "Finding relative hyperbolic structures", we provide a general algorithm that recognizes the class of groups that are hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in Israel J. Math, and Bull. London Math. Soc. respectivel