Related papers: A 3-manifold group which is not four dimensional l…

We classify closed, simply connected $n$-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions $2\leq n\leq 6$. In dimensions $3k$, $k=1,2$ there is only one such manifold…

In \cite{Chan95}, the authors classified the 2-extendable abelian Cayley graphs and posed the problem of characterizing all 2-extendable Cayley graphs. We first show that a connected bipartite Cayley (vertex-transitive) graph is…

Any two $(g; 0, 0, k)$-trisected closed 4-manifolds are related to each other by cutting and regluing in regard to $\gamma$ curves by an element of the Torelli group. Moreover, Lambert-Cole showed that this element can be replaced by an…

Let $G$ be a finite group and $\text{cd}(G)$ denote the character degree set for $G$. The prime graph $\Delta(G)$ is a simple graph whose vertex set consists of prime divisors of elements in $\text{cd}(G)$, denoted $\rho(G)$. Two primes…

Let $X$ be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry $\text{Nil}$. We show that the diffeomorphism group $\text{Diff}(X)$ deformation retracts to the isometry group $\text{Isom}(X)$. Combining this with…

We prove two results relating 3-manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3-manifold. If N has non-empty, toroidal boundary, and \pi_1(N) is a Kaehler group, then N is the…

A longstanding problem in rigidity theory is to characterize the graphs which are minimally generically rigid in 3-space. The results of Cauchy, Dehn, and Alexandrov give one important class: the triangulated convex spheres, but there is an…

Let $G$ be a connected reductive group defined over a non archimedean local field $k$. A theorem of Bernstein states that for any compact open subgroup $K$ of $G(k)$, there are, up to unramified twists, only finitely many $K$-spherical…

The concept of non-Gorenstein involutions on Calabi-Yau threefolds is a higher dimensional generalization of non-symplectic involutions on $K3$ surfaces. We present some elementary facts about Calabi-Yau threefolds with non-Gorenstein…

Let \(G\) be a finite group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). It is well known that, whenever \(\Delta(G)\) is connected, the diameter of…

We prove that the GKM graphs of GKM$_4$ manifolds that are either Hamiltonian or of complexity one extend to torus graphs. The arguments are based on a reformulation of the extension problem in terms of a natural representation of the…

A discussion of torsion of Riemannian G-structures leads to a survey of contributions of Alfred Gray and others on almost Hermitian manifolds, G_2-manifolds, curvature identities, volume expansions, plotting geodesics, and the geometry of…

In this note we give an example of a one-dimensional manifold with two connected components and a complete metric whose group of isometries has an orbit which is not closed. This answers a question of S. Gao and A. S. Kechris.

We consider groups defined by non-empty balanced presentations with the property that each relator is of the form R(x,y), where x and y are distinct generators and R(.,.) is determined by some fixed cyclically reduced word R(a,b) that…

A connected graph of order $n$ admitting a semiregular automorphism of order $n/k$ is called a $k$-multicirculant. Highly symmetric multicirculants of small valency have been extensively studied, and several classification results exist for…

We use differential forms on loop spaces to prove that the fundamental group of certain geometric transformation groups is infinite. Examples include both finite and infinite dimensional Lie groups. The finite dimensional examples are the…

Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially…

We investigate the complexity of finding an embedded non-orientable surface of Euler genus $g$ in a triangulated $3$-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance…

We give a very concise review of the group field theory formalism for non-perturbative quantum gravity, a higher dimensional generalisation of matrix models. We motivate it as a simplicial and local realisation of the idea of 3rd…

A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an…