Sorin Popa
We present here a shorter version of the proof of a result from our paper ``On a class of type II$_1$ factors with Betti numbers invariants'', showing that the von Neumann factor associated with the group $\Bbb Z^2 \rtimes SL(2, \Bbb Z)$ has trivial fundamental group.
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We propose to study some properties of the $C^*$-algebra naturally built out of the fundamental action that an automaton group $G$ admits on a regular rooted trees $\tree$.
In this paper we give a number of explicit constructions for II$_1$ factors and II$_1$ equivalence relations that have prescribed fundamental group and outer automorphism group. We construct factors and relations that have uncountable fundamental group different from $\IRpos$. In fact, given any II$_1$ equivalence relation, we construct a II$_1$ factor with the same fundamental group. Given any locally compact unimodular second countable group $G$, our construction gives a II$_1$ equivalence relation $\RelR$ whose outer automorphism group is $G$. The same construction does not give a II$_1$ fa…
In this paper we provide new examples of property (T) group factors with trivial fundamental group thereby providing more evidence towards Popa's conjecture on triviality of fundamental groups for property (T) group factors (page 9 \cite{Po13}; see also Problem 2, page 551 in Connes' book \cite{Co94}). Our groups arise as direct product of groups either in Class $\mathscr S$ or Class $\mathscr V$, as introduced in \cite{CDHK20}. We establish a unique prime factorization result for these product groups, thereby providing more evidence towards Popa's conjecture on prime decomposition of property…
We consider the realization of fundamental groups of $AF$-algebras in a certain class. We find the fundametal groups of $AF$-algebras with finite dimensional trace space which is not realizable as a fundamental group of von Neumann algebras.
Let $M$ be a $\rm II_1$ factor and let $\mathcal{F}(M)$ denote the fundamental group of $M$. In this article, we study the following property of $M$: for arbitrary $\rm II_1$ factor $B$, we have $\mathcal{F}(M \overline{\otimes} B)=\mathcal{F}(M)\mathcal{F}(B)$. We prove that for any subgroup $G\leq \mathbb{R}^*_+$ which is realized as a fundamental group of a $\rm II_1$ factor, there exists a $\rm II_1$ factor $M$ which satisfies this property and whose fundamental group is $G$. Using this, we deduce that if $G,H \leq \mathbb{R}^*_+$ are realized as fundamental groups of $\rm II_1$ factors (w…
We construct a group measure space II$_1$ factor that has two non-conjugate Cartan subalgebras. We show that the fundamental group of the II$_1$ factor is trivial, while the fundamental group of the equivalence relation associated with the second Cartan subalgebra is non-trivial. This is not absurd as the second Cartan inclusion is twisted by a 2-cocycle.
We introduce the fundamental group $rF(cA)$ of a unital $C^*$-algebra $cA$ with finite dimensional trace space. The elements of fundamental group are restricted by K-theoretical obstruction and positivity. Moreover we show there are uncountably many mutually nonisomorphic simple $C^{*}$-algebras such that $rF(cA)=\set{I_{n}}$. Our study is due to the results of fundamental group by Nawata and Watatani.
We introduce the fundamental group ${\mathcal F}(A)$ of a unital simple $C^*$-algebra $A$ with a unique normalized trace. We compute fundamental groups ${\mathcal F}(A)$ of several nuclear or non-nuclear $C^*$-algebras $A$. K-theoretical obstruction enables us to compute the fundamental group easily. Our study is essentially based on the computation of Picard groups by Kodaka.
We introduce the fundamental group ${\mathcal F}(\mathcal{R}_{G, ϕ})$ of a uniquely ergodic Cantor minimal $G$-system $\mathcal{R}_{G, ϕ}$ where $G$ is a countable discrete group. We compute fundamental groups of several uniquely ergodic Cantor minimal $G$-systems. We show that if $\mathcal{R}_{G, ϕ}$ arises from a free action $ϕ$ of a finitely generated abelian group, then there exists a unital countable subring $R$ of $\mathbb{R}$ such that $\mathcal{F}(\mathcal{R}_{G, ϕ})=R_{+}^\times$. We also consider the relation between fundamental groups of uniquely ergodic Cantor minimal $\mathbb{Z}^n…
We introduce the fundamental group $F(\mathcal{M})$ of a finite von Neumann algebra $\mathcal{M}$ with finite dimensional normal trace space. The form of $F(\mathcal{M})$ is completely determined. Moreover, there exists a finite von Neumann algebra $\mathcal{M}$ with finite dimensional normal trace space such that $F(\mathcal{M})=G$ for any conceivable groups $G$ as fundamental group.
We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about $k$-graphs when we show that the abelianisation of fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.
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