We study proper actions on homogeneous spaces of the group $\mathrm{B}(n)$ of invertible upper triangular matrices from a coarse-geometric viewpoint. We obtain sufficient conditions for the properness of the natural $L$-action on $\mathrm{B}(n)/H$, where $L,H$ are closed subgroups of $\mathrm{B}(n)$. We also give an explicit sufficient condition for properness of the $L$-action on a homogeneous space naturally identified with $\mathrm{B}(n)/H\cong \mathbb{R}^{n-1}$. In addition, we discuss extensions to the case where the ambient group $G$ is a subgroup of $\mathrm{B}(n)$ and $L,H$ are closed subgroups of $G$, with particular attention to the unipotent subgroup $\mathrm{N}(n)$.
Let $G$ be a group acting continuously on a compact metric space $X$. This naturally induces an action of $G$ on the space $\mathcal{K}(X)$ of nonempty compact subsets of $X$. In [KS26], Kra and Schmieding define a system $(X,G)$ to be IC-rigid if the union of the supports of the $G$-invariant Borel probability measure...
Adam Kanigowski, Kosma Kasprzak, Miriam Lorenzo-Laguno· 0 citations
Let $G$ be a connected compact Lie group and $R$ a commutative ring over which $H_*(G;R)$ is an exterior algebra on odd primitive generators. We prove a rigidity theorem: the Batalin-Vilkovisky (BV) algebra $\mathbb{H}_*(LG;R)$ of Chas and Sullivan on the homology of the free loop space is that of the product of odd sp...
Let $X$ be a closed, smooth, orientable, positive definite $4$-manifold with $H_1(X ; \mathbb{Z}) = 0$. Let $p$ be a prime and suppose that $G = \mathbb{Z}_p$ acts smoothly on $X$ (if $p=2$ we also require an assumption on how $G$ acts on $H^2(X ; \mathbb{Z})$). Using equivariant Yang--Mills theory, Hambleton--Lee and...
Let $Z$ be a compact K\"ahler manifold endowed with a Hamiltonian action of a compact connected Lie group $U$, and let $G\subset U^{\mathbb C}$ be a real compatible subgroup. Using the gradient map $\mu_{\mathfrak p}$ associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie alge...
Let $G$ be a complex reductive affine algebraic group with its symmetric tensor category $\mathcal{C}_G$ of finite-dimensional rational representations. We prove that every simple commutative algebra object with at most countable dimension in the direct limit completion $\operatorname{Ind}(\mathcal{C}_G)$ of $\mathcal{...
S. Carpi, Tiziano Gaudio, Luca Giorgetti· 0 citations
Let $\mathcal{G}(G,A,\alpha)$ be a discrete 2-group, given as the {categorical} extension of an ordinary group $G$ by an abelian group $A$. We define principal $\mathcal{G}$-bundles over a manifold in terms of \v{C}ech data over a good open cover. In the case of the manifold $S^2$, we calculate this bicategory explicit...
