Related papers: Shellable Complexes from Multicomplexes

Given a connected semisimple Lie group $G$ and an arithmetic subgroup $\Gamma$, it is well-known that each irreducible representation $\pi$ of $G$ occurs in the discrete spectrum $L^2_{\text{disc}}(\Gamma\backslash G)$ of…

Let $\Gamma$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_G\Gamma$ has the same Zariski closure as $\Ad G$. If $\alpha \colon \Gamma \to \GL_n(\real)$ is any finite-dimensional representation…

A discrete subgroup $\Gamma$ of a locally compact group $H$ is called a uniform lattice if the quotient $H/\Gamma$ is compact. Such an $H$ is called an envelope of $\Gamma$. In this paper we study the problem of classifying envelopes of…

Let $G$ be a connected and simply connected two-step nilpotent Lie group and $\Gamma$ a lattice subgroup of $G$. In this note, we give a new multiplicity formula, according to the sense of Moore, of irreducible unitary representations…

We prove that if \Gamma is subgroup of Diff_{+}^{1+\epsilon}(I) and N is a natural number such that every non-identity element of \Gamma has at most N fixed points then \Gamma is solvable. If in addition \Gamma is a subgroup of…

Let $G$ be a simply connected solvable Lie group with a lattice $\Gamma$ and $N$ the nilradical of $G$. For a complex valued representation $\rho: G\to GL(V_{\rho})$ such that the restriction $\rho_{|_{N}}$ is unipotent, as an advanced…

The value semigroup $\Gamma$ and the value set $\Lambda$ of $1$-forms are, respectively, a topological and an analytical invariant of a plane branch. Giving a plane branch $\mathcal{C}$ with semigroup $\Gamma$ there are a finitely number of…

For a discrete group $\Gamma$ satisfying some finiteness conditions we give a Bredon projective resolution of the trivial module in terms of projective covers of the chain complex associated to certain posets of subgroups. We use this to…

Let $\Gamma$ be a finite group and $V$ a finite-dimensional $\Gamma$-graded space over an algebraically closed field of characteristic not equal to 2. In the sense of conjugation, we classify all the so-called pre-nil or nil maximal abelian…

For a positive integer $k$ and a non-negative integer $t$ a class of simplicial complexes, to be denoted by $k$-${\rm CM}_t$, is introduced. This class generalizes two notions for simplicial complexes: being $k$-Cohen-Macaulay and…

The prime graph $\Gamma(G)$ of a finite group $G$ (also known as the Gruenberg-Kegel graph) has as its vertices the prime divisors of $|G|$, and $p\text-q$ is an edge in $\Gamma(G)$ if and only if $G$ has an element of order $pq$. Since…

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that…

Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S_D. This produces a collection of…

We study the cohomology $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$ of the deRham complex $\Lambda^*(G/\Gamma)\otimes{\mathbb C}$ of a compact solvmanifold $G/\Gamma$ with a deformed differential $d_{\lambda \omega}=d + \lambda\omega$,…

We study properly discontinuous and cocompact actions of a discrete subgroup $\Gamma$ of an algebraic group $G$ on a contractible algebraic manifold $X$. We suppose that this action comes from an algebraic action of $G$ on $X$ such that a…

Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact hypersurfaces in $\mathbb{R}^{n-1}$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $\Gamma\subset \mathbb{R}\mathrm{P}^n$ such…

We show that if $\Gamma = \Gamma_1\times\dotsb\times \Gamma_n$ is a product of $n\geq 2$ non-elementary ICC hyperbolic groups then any discrete group $\Lambda$ which is $W^*$-equivalent to $\Gamma$ decomposes as a $k$-fold direct sum…

Let $\Gamma$ be a finitely generated nilpotent group and let G be a complex reductive algebraic group. The representation variety $\mathrm{Hom}(\Gamma,G)$ and the character variety $\mathrm{Hom}(\Gamma,G)//G$ each carry a natural topology,…

We compute the Bieri-Neumann-Strebel invariants $\Sigma^1$ for the generalized solvable Baumslag-Solitar groups $\Gamma_n$ and their finite index subgroups. Using $\Sigma^1$, we show that certain finite index subgroups of $\Gamma_n$ cannot…

Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb R$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature.…