Narrow Results

In this paper, we show that every closed, locally homogeneous Riemannian manifold with positive simplicial volume must be homeomorphic to a locally symmetric space of non-compact type, giving a converse to a result by Lafont and Schmidt within the scope of closed, locally homogeneous Riemannian manifolds. This characterizes all closed locally homogeneous Riemannian manifolds with nonzero simplicial volume.

In Sternfeld’s work on Kolmogorov’s Superposition Theorem appeared the combinatorial–geometric notion of a basic set and a certain kind of arrays. A subset $X\subset {ℝ}^n$ is basic if any continuous function $X\to ℝ$ could be represented as the sum of compositions of continuous functions $ℝ\to ℝ$ and projections to the coordinate axes.

The definition of a Sternfeld array is presented in this paper.

Sternfeld’s Arrays Theorem.If a closed bounded subset$X\subset {ℝ}^{2n}$ contains Sternfeld arrays of arbitrary large size then$X$ is not basic.

The paper provides a simpler proof of this theorem.

This is a survey of some problems in geometric group theory that I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For each problem, I try to mention the author. If the author is not given, the problem, to the best of my knowledge, was formulated by me first.

Let $M$ be a graph manifold containing a single JSJ torus $T$ and whose JSJ blocks are of the form $\mathrm{\Sigma }\times S^1$, where $\mathrm{\Sigma }$ is a compact orientable surface with boundary. We show that if $M$ does not admit a Riemannian metric of everywhere nonpositive sectional curvature, then there is an essential curve on $T$ such that any finite-dimensional linear representation of ${\pi }_1(M)$ maps an element representing that curve to a matrix all of whose eigenvalues are roots of $1$. In particular, this shows that ${\pi }_1(M)$ does not admit a faithful finite-dimensional unitary representation, and gives a new proof that ${\pi }_1(M)$ is not linear over any field of positive characteristic.

In this paper, we discuss the relationship between conformal transformations of ${ℝ}^n\cup \{\infty \}$ and the curvature of curves. First, for any non-circular closed curve, there exists a length-preserving inversion such that the maximum pointwise absolute curvature can be made arbitrarily large. In contrast, we show that the total absolute curvatures of a family of curves conformally equivalent to a given simple or simple closed curve are uniformly bounded. Furthermore, we show that the total absolute curvature of an inverted regular $C^2$ simple closed curve as a function of inversion center and radius is removably discontinuous along the curve with exactly a $2\pi$ drop, and continuous elsewhere.

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.

An $n$-crossing projection of a link $L$ is a projection of $L$ onto a plane such that $n$ points on $L$ are superimposed on top of each other at every crossing. We prove that for all $k\in \mathbb{N}$ and all links $L$, the inequality

We consider two mod-p central series of the free group given by Stallings and Zassenhaus. Applying these series to definitions of Dennis Johnson's filtration of the mapping class group, we obtain two mod-p Johnson filtrations. Further, we adapt the definition of the Johnson homomorphisms to obtain mod-p Johnson homomorphisms.

We calculate the image of the first of these homomorphisms. We give generators for the kernels of these homomorphisms as well. We restrict the range of our mod-p Johnson homomorphisms using work of Morita. We finally prove the announced result of Perron that a rational homology 3-sphere may be given as a Heegaard splitting with gluing map coming from certain members of our mod-p Johnson filtrations.