Since the 1980s, the Bogomolov multiplier of a finite group has been known to obstruct rationality in complex algebraic geometry, and more recently it is understood to be responsible for any torsion in the oriented and stable unitary 2-dimensional $G$-equivariant bordism groups $\Omega_2^{SO,G}$ and $\Omega_2^{U,G}$. In this note, as a small step toward building a bridge between these two far-flung roles, we discuss the vanishing of Bogomolov multipliers of two specific families of finite groups. First, we revisit Kunyavski\u{i}'s result that the Bogomolov multipliers of all finite simple groups vanish, taking inspiration from the low-dimensional topological interpretation of the Ore conjecture. Second, in lieu of arguments in complex birational geometry (such as the hard direction of the Chevalley-Shephard-Todd theorem), we combine cut-and-paste combinatorial-topological techniques with elementary calculations of Ihara-Yokonuma to show that all finite Coxeter groups have vanishing Bogomolov multiplier.
In this paper, we address the following question: if a flat torus $\mathbb{T}^n$ is isometrically and minimally embedded into a sphere $\mathbb{S}^N$, must its translation group extend to the isometry group of the ambient sphere? As shown by Robert Bryant, for $n=2$ the answer is positive. Furthermore, while Ying Lu, Peng Wang, and Zhenxiao Xie recently demonstrated that the answer is negative for immersions when $n \geq 3$, the question for embeddings remained open. This problem is deeply tied to the work of Mikhail Gromov and Anton Petrunin concerning optimal curvature bounds. Petrunin proved that any immersion of a torus into a unit ball must have a maximum normal curvature of at least $\sqrt{\frac{3n}{n+2}}$. This bound is attained, for example, by families of tori constructed by Gromov. We call the tori that attain this optimal bound"Gromov tori". In this work, we first demonstrate that any Gromov torus is intrinsically flat, lies within a sphere, and is minimal inside it. We then establish the necessary and sufficient conditions for defining these tori. Finally, we present our main result: for dimensions $n \ge 3$, there exists a non-equivariant embedded Gromov torus, which provides a definitive negative answer to the question above.
We study left-invariant generalized complex structures on almost abelian Lie groups $G_A = \mathbb{R} \ltimes_{e^{tA}} \mathbb{R}^{d}$, and on the associated solvmanifolds. Our starting point is that the $\mathbf{i}$-eigenspace $\mathfrak{L}$ of such a structure is itself an almost abelian complex Lie algebra, a fact that governs everything that follows. When the structure matrix $A$ is diagonalizable over $\mathbb R$, we classify the types that occur in terms of the spectrum of $A$: they are prescribed by pairs of identical eigenvalues, pairs of opposite eigenvalues, and one remaining eigenvalue, the extremal types recovering the known classification of left-invariant complex and symplectic structures. We characterize the generalized Calabi--Yau case by a single linear condition on the eigenvalues, exhibit groups admitting only structures of intermediate type, and prove that admissible types come in adjacent pairs. Beyond the diagonalizable case we give an upper bound for the type in terms of the Jordan data of $A$, in dimension $6$, classify (non-extremal) left-invariant generalized complex structures, the first dimension in which these can occur. Finally, we compute the generalized Dolbeault cohomologies of the classified structures: it is given by a closed counting formula over sub-multisets of that spectrum, together with a duality in the grading.
Mauro de Andrade Pinto, Letícia Camponês Do Brasil Maia, Leonardo F. Cavenaghi et al.· 0 citations
Let $G$ be a torsion-free one-ended hyperbolic group and let $\phi\in\operatorname{Aut} (G)$. We prove that the mapping torus, or suspension, $$ M:=G_{\phi}=G\rtimes_\phi \mathbb{Z} $$ has solvable conjugacy problem. This builds on the pioneering work of Pr\'eaux who solved the conjugacy problem for all (geometrisable) three-manifolds. We view $M$ as a generalisation of a fibred three-manifold. Our proof begins with the canonical JSJ tree of $G$, whose suspension gives a graph-of-groups decomposition of $ M $. We refine each suspended QH vertex using a Nielsen--Thurston reduction system for its induced monodromy, taking account of non-orientable surfaces and orientation-reversing monodromy. Equivalently, this is a further geometric JSJ decomposition for each vertex which is a fibred three-manifold, but allowing Klein bottles as well as tori in the splitting. We then `block'pieces of this refined JSJ together according to whether they share a central element, glued along an elementary vertex, by a sequence of folding operations. This new splitting is cocompact and acylindrical; in particular certain local `solution sets'turn out to be rational subsets of virtually abelian subgroups - the virtually abelian groups in question are products of the edge groups. Our version of Pr\'eaux's graph-of-groups argument then reduces global conjugacy to effective intersection and non-emptiness for these rational sets.
In this paper we give a geometric description of the behavior of analytic intertwining operators between parabolically induced representations of $\mathrm{GL}_n(\mathrm{F})$, where $\mathrm{F}$ is a local non-archimedean field, in terms of hyperbolic localization functors of Braden. As a consequence, we can show that the image of an intertwining operator is always semi-simple and give a lower bound on the order of its pole, which is conjectured to be an equality as well as an upper bound in terms of the singular support of certain perverse sheaves. Moreover, we are able to reduce the conjecture of Lapid and M\'inguez on the shape of irreducible subrepresentations of induced representations to a computation of characteristic cycles. The theory of mixed Hodge modules plays a crucial role in the proofs.
Let $X$ carry a transversely holomorphic foliation, equivalently an elliptic involutive structure $V\subset T_{\mathbb C}X$, and let ${\mathcal O}_V$ be its sheaf of leafwise-constant, transversely holomorphic functions. We construct a finite superconnection model for the derived category of coherent ${\mathcal O}_V$-modules. The key input is a mixed local reduction for finite Maurer--Cartan objects over the mixed de Rham--Dolbeault dga $(\wedge^\bullet V^\vee,d_V)$: a multiplicative homotopy contracts the real directions, after which Block's Dolbeault gauge theorem removes the positive transverse form degrees. For compact $X$, this gives an exact equivalence between the homotopy category of bounded finite-rank flat $V$-superconnections and $D^b_{\mathrm{coh}}(X,{\mathcal O}_V)$, interpolating between the de Rham and Dolbeault realizations. We prove locally finite \v{C}ech descent under necessary uniform amplitude and rank bounds, identify the coherent heart with equivariant coherent analytic sheaves on a transverse monodromy groupoid, and characterize descent to ordinary holonomy. For a holomorphic suspension, we identify the full superconnection category, up to Morita equivalence, with the homotopy fixed points of the Dolbeault category of the transversal and derive an equivariant Ext spectral sequence. Examples on $S^1$ and $S^2$ delimit when ordinary monodromy $1$-groupoids can recover the derived category.
We study the monodromy action of the mixed braid group $B_{n,\mathcal{P}}$ on the first cohomology of cyclic branched covers of $\mathbb{P}^1$, which are mutually determined by a partition of branch points by equal ramification. The monodromy representation splits into irreducible representations on the $t$-eigenspaces of the deck transformation. For each, we construct an explicit spanning set using lifts of Pochhammer contours and figure-eight curves, and compute the Hermitian intersection form. The representation factors through a reduced mixed braid group by dropping $t$-invisible parts of the partition (those with trivial local monodromy). In this reduced representation, each generator acts by a complex reflection when the corresponding spanning class is non-isotropic, and by a unitary transvection when it is isotropic. Provided $\infty$ has non-trivial local monodromy, the factored representation is isomorphic to the reduced multivariate Burau representation evaluated at $t$.
V. AthiraE, Pranav Haridas· 0 citations
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