The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

When equipped with the natural topology first defined by Chabauty, the closed subgroups of a locally compact group $G$ form a compact space \Cal C(G). We analyse the structure of \Cal C(G) for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group $H$. We prove that \Cal C(H) is a 6-dimensional space that is path--connected but not locally connected. The lattices in $H$ form a dense open subset \Cal L(H) \subset \Cal C(H) that is the disjoint union of an infinite sequence of pairwise--homeomorphic aspherical manifolds of dimension six, each a torus bundle over $(\bold S^3 \smallsetminus T) \times \bold R$, where $T$ denotes a trefoil knot. The complement of \Cal L(H) in \Cal C(H) is also described explicitly. The subspace of \Cal C(H) consisting of subgroups that contain the centre $Z(H)$ is homeomorphic to the 4--sphere, and we prove that this is a weak retract of \Cal C(H).Comment: Minor edits. Final version. To appear in the Pacific Journal. 41 pages, no figure

On the Margulis constant for Kleinian groups, I curvature

The Margulis constant for Kleinian groups is the smallest constant $c$ such that for each discrete group $G$ and each point $x$ in the upper half space ${\bold H}^3$, the group generated by the elements in $G$ which move $x$ less than distance c is elementary. We take a first step towards determining this constant by proving that if $\langle f,g \rangle$ is nonelementary and discrete with $f$ parabolic or elliptic of order $n \geq 3$, then every point $x$ in ${\bold H}^3$ is moved at least distance $c$ by $f$ or $g$ where $c=.1829\ldots$. This bound is sharp

Curves having one place at infinity and linear systems on rational surfaces

Denoting by ${\mathcal L}_d(m_0,m_1,...,m_r)$ the linear system of plane curves passing through $r+1$ generic points $p_0,p_1,...,p_r$ of the projective plane with multiplicity $m_i$ (or larger) at each $p_i$, we prove the Harbourne-Hirschowitz Conjecture for linear systems ${\mathcal L}_d(m_0,m_1,...,m_r)$ determined by a wide family of systems of multiplicities $\bold{m}=(m_i)_{i=0}^r$ and arbitrary degree $d$. Moreover, we provide an algorithm for computing a bound of the regularity of an arbitrary system $\bold{m}$ and we give its exact value when $\bold{m}$ is in the above family. To do that, we prove an $H^1$-vanishing theorem for line bundles on surfaces associated with some pencils ``at infinity''.Comment: This is a revised version of a preprint of 200

Quantum-classical phase transition of the escape rate of two-sublattice antiferromagnetic large spins

The Hamiltonian of a two-sublattice antiferromagnetic spins, with single (hard-axis) and double ion anisotropies described by $H=J \bold{\hat{S}}_{1}\cdot\bold{\hat{S}}_{2} - 2J_{z}\hat{S}_{1z}\hat{S}_{2z}+K(\hat{S}_{1z}^2 +\hat{S}_{2z}^2)$ is investigated using the method of effective potential. The problem is mapped to a single particle quantum-mechanical Hamiltonian in terms of the relative coordinate and reduced mass. We study the quantum-classical phase transition of the escape rate of this model. We show that the first-order phase transition for this model sets in at the critical value $J_c=\frac{K+J_z}{2}$ while for the anisotropic Heisenberg coupling $H = J(S_{1x}S_{2x} +S_{1y}S_{2y}) + J_zS_{1z}S_{2z} + K(S_{1z}^2+ S_{2z}^2)$ we obtain $J_c=\frac{2K-J_z}{3}$ . The phase diagrams of the transition are also studied.Comment: 7 pages, 3 figure

Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and \break Kadomtsev-Petviashvili (KP) equations were constructed for a given curve $y^2 = f(x)$ whose genus is three. This study was based upon the fact that about one hundred years ago (Acta Math. (1903) {\bf{27}}, 135-156), H. F. Baker essentially derived KdV hierarchy and KP equation by using bilinear differential operator ${\bold{D}}$, identities of Pfaffians, symmetric functions, hyperelliptic $\sigma$-function and $\wp$-functions; $\wp_{\mu \nu} = -\partial_\mu \partial_\nu \log \sigma$ $= - ({\bold{D}}_\mu {\bold{D}}_\nu \sigma \sigma)/2\sigma^2$. The connection between his theory and the modern soliton theory was also discussed.Comment: AMS-Tex, 12 page

The freudenthal space for approximate systems of compacta and some applications

In this paper we define a space $\sigma (\underline{\bold X})$ for approximate systems of compact spaces. The construction is due to H. Freudenthal for usual inverse sequences \cite{4, p. 153--156}. We stablish the following properties of this space: (1) The space $\sigma(\underline{\bold X})$ is a paracompact space, (2) Moreover, if $\underline{\bold X}$ is an approximate sequence of compact (metric) spaces, then $\sigma(\underline{\bold X})$ is a compact (metric) space (Lemma 2.4). We give the following applications of the space $\sigma(\underline{\bold X})$: (3) If $\underline{\bold X}$ is an approximate system of continua, then $X=\lim \underline{\bold X}$ is a continuum (Theorem 3.1), (4) If $\underline{\bold X}$ is an approximate system of hereditarily unicoherent spaces, then $X=\lim \underline{\bold X}$ is hereditarily unicoherent (Theorem 3.6), (5) If $\underline{\bold X}$ is an approximate system of trees with monotone onto bonding mappings, then $X=\lim\underline{\bold X}$ is a tree (Theorem 3.13)