Universal groups of intermediate growth and their invariant random subgroups

We exhibit examples of groups of intermediate growth with $2^{\aleph_0}$ ergodic, continuous, invariant random subgroups. The examples are the universal groups associated with a family of groups of intermediate growth.Comment: Functional Analysis and its Applications, 201

Integrating Faith and Learning: Preparing Teacher Candidates to Serve Culturally and Linguistically Diverse Students

This essay examines how liberation theology and critical pedagogy inform the integration of faith and learning of a teacher educator who felt called to serve culturally and linguistically diverse students in the United States. The essay provides a brief cultural background of the educator’s journey from instructional assistant in an English learner program to teacher educator at a Christian University. The essay explains how liberation theology and critical pedagogy provide a coherent framework for preparing teacher candidates to work with English learners in public schools

Small spectral radius and percolation constants on non-amenable Cayley graphs

Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $\Gamma$, does there exist a generating set $S$ such that the Cayley graph $(\Gamma,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(\Gamma,S)<p_u(\Gamma,S)$? We show that this is true if $\Gamma$ contains an infinite normal subgroup $N$ such that $\Gamma/ N$ is non-amenable. Moreover for any finitely generated group $G$ containing $\Gamma$ there exists a generating set $S'$ of $G$ such that $p_c(G,S')<p_u(G,S')$. In particular this applies to free Burnside groups $B(n,p)$ with $n \geq 2, p \geq 665$. We also explore how various non-amenability numerics, such as the isoperimetric constant and the spectral radius, behave on various growing generating sets in the group

Set-theoretic solutions of the Yang-Baxter equation, Braces, and Symmetric groups

We involve simultaneously the theory of matched pairs of groups and the theory of braces to study set-theoretic solutions of the Yang-Baxter equation (YBE). We show the intimate relation between the notions of a symmetric group (a braided involutive group) and a left brace, and find new results on symmetric groups of finite multipermutation level and the corresponding braces. We introduce a new invariant of a symmetric group $(G,r)$, \emph{the derived chain of ideals of} $G$, which gives a precise information about the recursive process of retraction of $G$. We prove that every symmetric group $(G,r)$ of finite multipermutation level $m$ is a solvable group of solvable length at most $m$. To each set-theoretic solution $(X,r)$ of YBE we associate two invariant sequences of symmetric groups: (i) the sequence of its derived symmetric groups; (ii) the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction. We find new criteria necessary and sufficient to claim that $(X, r)$ is a multipermutation solution.Comment: 44 page

On distinguishability, orthogonality, and violations of the second law: contradictory assumptions, contrasting pieces of knowledge

Two statements by von Neumann and a thought-experiment by Peres prompts a discussion on the notions of one-shot distinguishability, orthogonality, semi-permeable diaphragm, and their thermodynamic implications. In the first part of the paper, these concepts are defined and discussed, and it is explained that one-shot distinguishability and orthogonality are contradictory assumptions, from which one cannot rigorously draw any conclusion, concerning e.g. violations of the second law of thermodynamics. In the second part, we analyse what happens when these contradictory assumptions comes, instead, from _two_ different observers, having different pieces of knowledge about a given physical situation, and using incompatible density matrices to describe it.Comment: LaTeX2e/RevTeX4, 18 pages, 6 figures. V2: Important revisio