Palindromic primitives and palindromic bases in the free group of rank two

The present paper records more details of the relationship between primitive elements and palindromes in F_2, the free group of rank two. We characterise the conjugacy classes of primitive elements which contain palindromes as those which contain cyclically reduced words of odd length. We identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length. We show that under obvious conditions on exponent sums, pairs of palindromic primitives form palindromic bases for F_2. Further, we note that each cyclically reduced primitive element is either a palindrome, or the concatenation of two palindromes.Comment: 8 page

Rigidity of graph products of abelian groups

We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.Comment: 11 pages, 1 figur

The Bieri-Neumann-Strebel Invariant of the Pure Symmetric Automorphisms of a Right-Angled Artin Group

We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links

On the automorphisms of a graph product of abelian groups

We study the automorphisms of a graph product of finitely-generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition of Aut* W in which one of the factors is Inn W. We also give a number of applications, some of which are geometric in nature.Comment: 38 pages, 4 figure

AUTHENTIC ASSESSMENT TO ENGAGE STUDENTS AND EQUIP THEM WITH MODERN SKILLS

The ACSME 2020 theme reminds us that science graduates, including mathematics graduates, need broad skills so that they can contribute meaningfully to solving important problems and adapt quickly in the face of uncertainty. It is not enough that a graduate can solve any problem in the textbook, they need to see the potential ways that their skills can contribute to the big picture, and they must be able to articulate this potential to non-mathematicians. In a way, they must be storytellers. Well-designed authentic assessment tasks, as championed by Wiggins since the 1980s, ask students to solve problems that seem real and a little messy. They allow students to do work that seems personal, and of which they are proud. This allows students to see the value of their discipline, and to practice the many competencies that graduates need but we often do not explicitly teach or assess. Authentic assessment tasks can be implemented in courses large and small, and at any year-level. In this workshop, you will develop an idea for an authentic assessment item in an undergraduate mathematics or science course, and a marking criteria for the assessment. The goal is to leave with a plan for a task that you can implement in a course that you teach and a marking criteria that encourages students to buy-in to the task in the way that you want them to. To guide you, we will describe examples of authentic assessment tasks, and the corresponding marking criteria, that have proved effective for undergraduates in mathematics. These examples have been tried and tested by the presenter over 13 years teaching undergraduate mathematics in the liberal arts setting in the USA, and two years teaching quantitative material to science students in a large course for first-semester first-year science students at the University of Queensland (UQ). Please find attached one of the example tasks and the marking criteria. This task was developed by the presenter for a large first-year interdisciplinary science course compulsory at UQ. In each semester since it was implemented, a majority of students reported that the assignment itself had a positive impact on their attitudes to and perceptions of science, mathematics and computer programming

Formative authentic assessment to develop communication competencies among first-year science students

BACKGROUND SCIE1100 Advanced Theory and Practice in Science is a first-year course taken by approximately 100 high-achieving students in the Bachelor of Advanced Science (Honours) program at the University of Queensland.  The learning objectives of the course include objectives related to critical thinking, understanding science within a societal context, and communication competencies. Course feedback in 2019 suggested that SCIE1100 did not provide sufficient challenge for some students, and it did not provide enough explicit instruction in communication competencies. AIMS This research aims to evaluate the effectiveness of a course redesign undertaken in 2020. DESCRIPTION OF INTERVENTION For Semester 1, 2020, the author redesigned SCIE1100 in order to better develop communication competencies.   The redesign combined several well-known pedagogical principles: constructive alignment, formative assessment, authentic assessment and criteria-referenced assessment.  A key component of the redesign was a sequence of ten formative authentic assessment tasks. DESIGN AND METHODS The course redesign was evaluated using mixed methods.  Aggregated student grades in the ten tasks were analyzed for trends indicative of effective formative assessment; the performance of the 2020 cohort (N = 93) on a summative communication assessment was compared to the performance of the 2019 cohort (N =127) on the same task, graded by the same graders using the same marking criteria; mid-way through the semester, a student-led team conducted surveys (N = 37) and focus groups to evaluate students’ attitudes and experiences in the course; in the second last week of the semester, students completed the UQ Employability Framework Activity, and qualitative responses provided in this activity were examined for evidence of student self-efficacy and engagement. RESULTS Quantitatively, we observe a small positive effect in skill development; qualitatively, some students reported an improvement in self-efficacy and engagement, and some students reported spending more time on the tasks than the design intended. CONCLUSIONS The redesign succeeded in better delivering the learning outcomes related to communication competencies, possibly at the expense of over-working some students

The automorphism group of the free group of rank two is a CAT(0) group

We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.Comment: 7 pages, 2 figures. The manuscript has been modified in minor ways in accordance with a referee's recommendations, and a misattribution of the result "Aut F_2 is biautomatic" has been correcte