Related papers: Some free entropy dimension inequalities for subfa…

Let G be a group of the form G_1* ... *G_n, the free product of n subgroups, and let M be a ZG-module of the form $\bigoplus_{i=1}^n M_i \otimes_{\mathbb{Z}G_i} \mathbb{Z}G$. We shall give formulae in various situations for $d_{ZG}(M)$, the…

This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the first free entropy proof of Popa's famous result that the generator MASA in a…

We show that for every $N\ge 3$ the free unitary group $U^+_N$ is topologically generated by its classical counterpart $U_N$ and the lower-rank $U^+_{N-1}$. This allows for a uniform inductive proof that a number of finiteness properties,…

Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $\sum_{p\in \pi(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of…

In a statistical ensemble with $M$ microstates, we introduce an $M \times M$ correlation matrix with the correlations between microstates as its elements. Using eigenvectors of the correlation matrix, we can define eigen microstates of the…

A subgroup $\Delta\leq \Gamma$ is commensurated if $|\Delta:\Delta\cap \gamma\Delta\gamma^{-1}|<\infty$ for all $\gamma\in \Gamma$. We show a finitely generated branch group is just infinite if and only if every commensurated subgroup is…

In this paper, we extend the notion of non-microstate free entropy to the bi-free setting. Using a diagrammatic approach involving bi-non-crossing diagrams, bi-free difference quotients are constructed as analogues of the free partial…

For a topological space $X$ we propose to call a subset $S \subset X$ "free in $X$" if it admits a well-ordering that turns it into a free sequence in $X$. The well-known cardinal function $F(X)$ is then definable as $\sup\{|S| : S \text{…

We show that a free product of a II_1-factor and a finite von Neumann algebra with amalgamation over a finite dimensional subalgebra is always a II_1-factor, and provide an algorithm for describing it in terms of free products (with…

We introduce a modification of Voiculescu's free entropy which coincides with the liminf variant of Voiculescu's free entropy on extremal states, but is a concave upper semi-continuous function on the trace state space. We also extend the…

Let $X$ be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$ \mathbf{E}\|X\|_{S_p} \asymp \mathbf{E}\Bigg[ \Bigg(\sum_i\Bigg(\sum_j X_{ij}^2\Bigg)^{p/2}\Bigg)^{1/p} \Bigg]…

We study the entanglement entropy of random partitions in one- and two-dimensional critical fermionic systems. In an infinite system we consider a finite, connected (hypercubic) domain of linear extent $L$, the points of which with…

Let $M_0 \subset M_1$ be a finite-index infinite-depth hyperfinite $II_1$ subfactor and $\omega$ a free ultrafilter of the natural numbers. We show that if this subfactor is constructed from a commuting square then the central sequence…

We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all…

Suppose $M$ is a von Neumann algebra equipped with a faithful normal state $\varphi$ and generated by a finite set $G=G^*$, $|G|\geq 2$. We show that if $G$ consists of eigenvectors of the modular operator $\Delta_\varphi$ with finite free…

We describe groups elementarily equivalent to a free metabelian group with n generators. We also explore an exponentiation that naturally occurs in metabelian groups.

This paper is an exposition, with some new applications, of our results on the growth of entropy of convolutions. We explain the main result on $\mathbb{R}$, and derive, via a linearization argument, an analogous result for the action of…

We give a criterion for a set of $n$ hyperbolic isometries of a $\mathrm{CAT}(0)$ metric space $X$ to generate a free group on $n$ generators. This extends a result by Alperin, Farb and Noskov who proved this for 2 generators under the…

Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the…

Given an irreducible subshift of finite type X, a subshift Y, a factor map \pi : X \to Y, and an ergodic invariant measure \nu on Y, there can exist more than one ergodic measure on X which projects to \nu and has maximal entropy among all…