Solution of the Conjugacy Problem and Malnormality of Subgroups in Certain Relative Small Cancellation Group Presentations

2007 ◽  
pp. 121-168
Author(s):  
Arye Juhász
1996 ◽  
Vol 06 (04) ◽  
pp. 429-440 ◽  
Author(s):  
KARIN JOHNSGARD

Gersten and Short showed that group presentations satisfying certain minimal small cancellation conditions are automatic. The automatic structure they described does not have unique group element representatives. It is easy, however, to modify their language to produce an automatic spanning tree of geodesics: that is, an automatic structure of geodesics which is both prefix-closed and has unique representatives. When combined with a construction showing that all geodesic representatives of a group element are encoded in a tiling of a particular geometric form, this shows that in fact the language of lexicographically least geodesics is also automatic (it is clearly a spanning tree).


2007 ◽  
Vol 16 (09) ◽  
pp. 1219-1233 ◽  
Author(s):  
PAOLO BELLINGERI ◽  
EDDY GODELLE

We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.


1987 ◽  
Vol 102 (3) ◽  
pp. 443-451 ◽  
Author(s):  
Mohamed S. El-Mosalamy ◽  
Stephen J. Pride

This paper is concerned with group presentations satisfying the small cancellation condition T(6). The definition of this condition is given in §1·2, together with some examples. Before giving the definition, however, we describe (in §1·1) some material which, to a certain extent, motivated our paper. In § 1·3 we state our main theorem, which provides new solutions to the word and conjugacy problems for T(6)-groups.


Author(s):  
Martín Axel Blufstein ◽  
Elías Gabriel Minian ◽  
Iván Sadofschi Costa

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$ -groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$ , the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$ , which implies hyperbolicity.


2020 ◽  
Vol volume 12, issue 2 ◽  
Author(s):  
Alex Bishop ◽  
Michal Ferov

Small cancellation groups form an interesting class with many desirable properties. It is a well-known fact that small cancellation groups are generic; however, all previously known results of their genericity are asymptotic and provide no information about "small" group presentations. In this note, we give closed-form formulas for both lower and upper bounds on the density of small cancellation presentations, and compare our results with experimental data. Comment: 18 pages, 12 figures


2020 ◽  
Vol 7 (Supplement_1) ◽  
pp. S592-S593
Author(s):  
Lucille K Kohlenberg ◽  
Muriel J Solberg ◽  
Fatima N Ali-Mirza ◽  
Sheela Shenoi ◽  
Saad Omer

Abstract Background The COVID-19 pandemic has posed a unique challenge to undergraduate medical education. Medical schools postponed student participation in direct patient care in mid-March 2020, creating the need for rapidly-designed, virtual, and innovative learning experiences. Methods Utilizing Kern’s six-step approach to curriculum development, faculty and medical student liaisons rapidly designed a six-week online and interactive course for clerkship-year students and above, launched on March 30th, 2020. “Patients, Populations, and Pandemics: Responding to COVID-19” emphasized honing higher level skills of Bloom’s taxonomy, namely evaluating, synthesizing, and creating. Following weekly faculty-led lectures, student groups identified research questions, analyzed literature, presented data, critiqued peer presentations, and created infographics for the public. Results We aimed to maintain quality and interactiveness despite challenges posed by our timeframe, the evolving COVID-19 literature, and the virtual setting. We recruited frontline faculty and designed the course to facilitate discussion, thereby promoting real-time exploration of public health and clinical challenges. Encouraging student participation, we incorporated group synthesis sessions and instructed use of video, hand-raising, and chat features. In a survey administered at the end of the first week, 85.7% (18/21) of students strongly agreed or agreed that small group presentations successfully enabled synthesis of new and emerging data. Among the 29 enrolled students, 82.8% (24/29) of students completed final course evaluations, with 87.5% (21/24) agreeing that the learning activities “usually” or “always” helped meet the learning objectives identified at the beginning of the course. The course was rated as “excellent” or “very good” by 83.3% (20/24) of students. Conclusion Lessons learned include providing students with increased direction on critically reviewing peer presentations and imparting guidance on best practices for data synthesis. This course model will be disseminated throughout our institution and beyond to address challenges in remote learning and to serve as a paradigm during future health crises. Disclosures All Authors: No reported disclosures


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