Related papers: Maximal Haagerup subgroups in $\mathbb{Z}^{n+1}\rt…

We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $\mathbb{Z}^2 \rtimes SL_2(\mathbb{Z})$ and obtain several explicit instances…

We prove that $L(SL_2(\textbf{k}))$ is a maximal Haagerup von Neumann subalgebra in $L(\textbf{k}^2\rtimes SL_2(\textbf{k}))$ for $\textbf{k}=\mathbb{Q}$. Then we show how to modify the proof to handle $\textbf{k}=\mathbb{Z}$. The key step…

Let $n\geq 2$ and $G_n=\mathbb{Z}^n\rtimes SL_n(\mathbb{Z})$. We classify all $G_n$-invariant von Neumann subalgebras in $L(G_n)$. For $n=2$, this gives an alternative proof of the previous result of Jiang-Liu. For $n\geq 3$, this gives the…

We classify certain $\mathbb{Z}_2 $-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including:…

The unique irreducible representation of $\SL_2(\R)$ on $\R^n$ induces an action, called the \textit{linear action}, of $\SL_2(\Z)$ on the torus $\T^n$ for every $n\geq 2$. For $n$ odd, it factors through $\PSL_2(\Z)$, so we denote by $G_n$…

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and…

With every nontrivial connected algebraic group $G$ we associate a positive integer ${\rm gtd}(G)$ called the generic transitivity degree of $G$ and equal to the maximal $n$ such that there is a nontrivial action of $G$ on an irreducible…

We classify up to conjugation by $\operatorname{GL}(2,\mathbb{R})$ (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of $\operatorname{Sp}(2,\mathbb{R})$ by means of an essentially…

Subgroups $H_1$ and $H_2$ of a group $G$ are said to be locally conjugate if there is a bijection $f: H_1 \rightarrow H_2$ such that $h$ and $f(h)$ are conjugate in $G$ for every $h \in H_1$. This paper studies local conjugacy among…

Let $\F_q$ be the finite field with $q$ elements, $p=\Char \F_q$. The group $\GL_2(\F_q)$ acts naturally in the set of irreducible polynomials over $\F_q$ of degree at least $2$. In this paper we are interested in the characterization and…

Let $G$ and $N$ be finite groups of order $2n$ where $n$ is odd. We say the pair $(G,N)$ is Hopf-Galois realizable if $G$ is a regular subgroup of $\h(N)=N\rtimes\au(N)$. In this article we give necessary conditions on $G$ (similarly $N$)…

We classify all $G$-invariant von Neumann subalgebras in $L(G)$ for $G=\mathbb{Z}^2\rtimes SL_2(\mathbb{Z})$. This is the first result on classifying $G$-invariant von Neumann subalgebras in $L(G)$ for i.c.c. groups $G$ without the…

We show that for every mixing orthogonal representation $\pi : \Z \to \mathcal O(H_\R)$, the abelian subalgebra $\LL(\Z)$ is maximal amenable in the crossed product ${\rm II}_1$ factor $\Gamma(H_\R)\dpr \rtimes_\pi \Z$ associated with the…

We establish the existence of maximal subgroups of various diferent natures in SL(n,Z). In particular, we prove that there are continuously many maximal subgroups, we provide a maximal subgroup whose action on the projective space has no…

We prove that for any infinite, maximal amenable subgroup $H$ in a hyperbolic group $G$, the von Neumann subalgebra $LH$ is maximal amenable inside $LG$. It provides many new, explicit examples of maximal amenable subalgebras in II$_1$…

Maximal connected grading classes of a finite-dimensional algebra $A$ are in one-to-one correspondence with Galois covering classes of $A$ which admit no proper Galois covering and therefore are key in computing the intrinsic fundamental…

This paper investigates the maximal subgroups of a free projection-generated regular $*$-semigroup $PG(P)$ over a projection algebra $P$, and their relationship to the maximal subgroups of the free idempotent-generated semigroup $IG(E)$…

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, ..., x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle$. So $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z}…

Let $\boldsymbol{G}$ be an algebraic group of exceptional Lie type in characteristic $p$, $G=\boldsymbol{G}^{\sigma}$ its fixed-point subgroup under the action of a Steinberg endomorphism $\sigma$, and $\overline{G}$ an almost simple group…

In 1933 B.~H.~Neumann constructed uncountably many subgroups of ${\rm SL}_2(\mathbb Z)$ which act regularly on the primitive elements of $\mathbb Z^2$. As pointed out by Magnus, their images in the modular group ${\rm PSL}_2(\mathbb Z)\cong…