Height in splittings of hyperbolic groups

Suppose $H$ is a hyperbolic subgroup of a hyperbolic group $G$. Assume there exists $n > 0$ such that the intersection of $n$ essentially distinct conjugates of $H$ is always finite. Further assume $G$ splits over $H$ with hyperbolic vertex and edge groups and the two inclusions of $H$ are quasi-isometric embeddings. Then $H$ is quasiconvex in $G$. This answers a question of Swarup and provides a partial converse to the main theorem of \cite{GMRS}.Comment: 16 pages, no figures, no table

Thurston boundary of Teichm\"uller spaces and the commensurability modular group

If $p : Y \to X$ is an unramified covering map between two compact oriented surfaces of genus at least two, then it is proved that the embedding map, corresponding to $p$, from the Teichm\"uller space ${\cal T}(X)$, for $X$, to ${\cal T}(Y)$ actually extends to an embedding between the Thurston compactification of the two Teichm\"uller spaces. Using this result, an inductive limit of Thurston compactified Teichm\"uller spaces has been constructed, where the index for the inductive limit runs over all possible finite unramified coverings of a fixed compact oriented surface of genus at least two. This inductive limit contains the inductive limit of Teichm\"uller spaces, constructed in \cite{BNS}, as a subset. The universal commensurability modular group, which was constructed in \cite{BNS}, has a natural action on the inductive limit of Teichm\"uller spaces. It is proved here that this action of the universal commensurability modular group extends continuously to the inductive limit of Thurston compactified Teichm\"uller spaces.Comment: AMSLaTex file. To appear in Conformal Geometry and Dynamic

On a theorem of Scott and Swarup

Let 1 → H → G → Z → 1 be an exact sequence of hyperbolic groups induced by an automorphism Φ of the free group H. Let H1(⊂ H) be a finitely generated distorted subgroup of G. Then there exist N > 0 and a free factor K of H such that the conjugacy class of K is preserved by ΦN and H1 contains a finite index subgroup of a conjugate of K. This is an analog of a Theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds

Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.Comment: 27pgs No figs, v3: final version, incorporating referee's comments, to appear in Geometriae Dedicat

Molecular transistor coupled to phonons and Luttinger-liquid leads

We study the effects of electron-phonon interactions on the transport properties of a molecular quantum dot coupled to two Luttinger-liquid leads. In particular, we investigate the effects on the steady state current and DC noise characteristics. We consider both equilibrated and unequilibrated on-dot phonons. The density matrix formalism is applied in the high temperature approximation and the resulting semi-classical rate equation is numerically solved for various strengths of electron-electron interactions in the leads and electron-phonon coupling. The current and the noise are in general smeared out and suppressed due to intralead electron interaction. On the other hand, the Fano factor, which measures the noise normalized by the current, is more enhanced as the intralead interaction becomes stronger. As the electron-phonon coupling becomes greater than order one, the Fano factor exhibits super-Poissonian behaviour.Comment: 11 pages, 11 figure

Phonon runaway in nanotube quantum dots

We explore electronic transport in a nanotube quantum dot strongly coupled with vibrations and weakly with leads and the thermal environment. We show that the recent observation of anomalous conductance signatures in single-walled carbon nanotube (SWCNT) quantum dots can be understood quantitatively in terms of current driven `hot phonons' that are strongly correlated with electrons. Using rate equations in the many-body configuration space for the joint electron-phonon distribution, we argue that the variations are indicative of strong electron-phonon coupling requiring an analysis beyond the traditional uncorrelated phonon-assisted transport (Tien-Gordon) approach.Comment: 8 pages, 6 figure