Related papers: Presentations for the Euclidean Picard modular gro…

We show that several families of asymptotically rigid mapping class groups arise as explicit quotients of the fundamental group of a graph of groups, with mapping class groups as vertex and edge stabilizers. Using this description, and…

For a compact real form $U$ of a complex simple Lie group $G$, and an irreducible representation $\rho:\Gamma \to U$ of a Fuchsian group of the first kind $\Gamma$, it is shown that the classical isomorphism of Shimura, for the periods of a…

Let $(\mathbf{U}, \mathbf{U}^\imath)$ be a quasi-split quantum symmetric pair of Kac-Moody type. The $\imath$quantum group $\mathbf{U}^\imath$ admits a Serre presentation featuring the $\imath$-Serre relations in terms of $\imath$-divided…

A $p$-arithmetic subgroup of $\mathrm{SL}_2(\mathbb{Q})$ like the Ihara group $\Gamma := \mathrm{SL}_2(\mathbb{Z}[1/p])$ acts by M\"obius transformations on the Poincar\'e upper half plane $\mathcal{H}$ and on Drinfeld's $p$-adic upper half…

For a smooth geometrically integral variety $X$ over a field $k$ of characteristic 0, we introduce and investigate the extended Picard complex $UPic(X)$. It is a certain complex of Galois modules of length 2, whose zeroth cohomology is…

We study representations of lattices of PU(m,1) into PU(n,1). We show that if a representation is reductive and if m is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic m-space to complex…

We explicitly compute the ellitpic points and isotropy groups for the action of the Picard modular group over the Gaussian integers on 2-dimensional complex hyperbolic space.

Using a theorem proved by Bekka and Driutti, we show that if $\mathfrak{f}$ is a freely generated nilpotent Lie algebra of step-two, then almost every irreducible representation of the corresponding Lie group restricted to some lattice…

We consider a certain hybridization construction which produces a subgroup of ${\rm PU}(n,1)$ from a pair of lattices in ${\rm PU}(n-1,1)$. Among the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$, we show that the hybrid of pairs of…

We present a general framework for Matrix theory compactified on a quotient space R^n/G, with G a discrete group of Euclidean motions in R^n. The general solution to the quotient conditions gives a gauge theory on a noncommutative space. We…

In this note, we construct canonical bases for the spaces of weakly holomorphic modular forms with poles supported at the cusp $\infty$ for $\Gamma_{0}(4)$ of integral weight $k$ for $k\leq-1$, and we make use of the basis elements for the…

The question whether non-isomorphic finite $p$-groups can have isomorphic modular group algebras was recently answered in the negative by Garc\'ia-Lucas, Margolis and del R\'io [J. Reine Angew. Math. 783 (2022), pp. 269-274]. We embed these…

The orbits in $\Gamma_{\infty}(3) \backslash \Gamma(3)$ are in bijection with sets of invariants satisfying certain relations. We explain how wedge product matrices give an alternative definition of the invariants of matrix orbits. This new…

Let $X=S\times E \times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. Let $\Gamma$ be a discrete group acting isometrically and cocompactly on $X$. We…

We use the formalism of the Bergman tau functions to study the geometry of moduli spaces of holomorphic quadratic differentials on complex algebraic curves. We introduce two natural tau functions and interpret them as holomorphic sections…

One particular approach to quantum groups (matrix pseudo groups) provides the Manin quantum plane. Assuming an appropriate set of non-commuting variables spanning linearly a representation space one is able to show that the endomorphisms on…

We have recently proved a homological stability theorem for moduli spaces of r-Spin Riemann surfaces, which in particular implies a Madsen--Weiss theorem for these moduli spaces. This allows us to effectively study their stable cohomology,…

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar\'e equations on Lie groups and homogeneous spaces. Orbit…

A graphical expansion formula for non-commutative matrix integrals with values in a finite-dimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their non-orientable counterpart called Moebius graphs. The…

In this work, we show that extending the standard description of space-time symmetries from groups of isometries to the more flexible framework of kinematical groupoids allows for the extension of Wigner's program to curved space-times. We…