ar X iv : m at h / 01 05 23 2 v 1 [ m at h . N T ] 2 8 M ay 2 00 1 MODULAR CURVES OF GENUS 2

ar X iv : m at h / 05 05 24 4 v 1 [ m at h . G T ] 1 2 M ay 2 00 5 WORD HYPERBOLIC EXTENSIONS OF SURFACE GROUPS

Let S be a closed surface of genus g ≥ 2. A finitely generated group Γ S is an extension of the fundamental group π 1 (S) of S if π 1 (S) is a normal subgroup of Γ S. We show that the group Γ S is hyperbolic if and only if the orbit map for the action of the quotient group Γ = Γ S /π 1 (S) on the complex of curves is a quasi-isometric embedding.

ar X iv : m at h / 04 01 05 6 v 1 [ m at h . G T ] 7 J an 2 00 4 SQUARE - TILED SURFACES IN H ( 2 )

This is a study of square-tiled translation surfaces in the stratum H(2) and their SL(2, R)-orbits or Teichmüller discs, which are arithmetic. We prove that for prime n > 3 translation surfaces tiled by n squares fall into two Teichmüller discs, only one of them with elliptic points, and that the genus of these discs has a cubic growth rate in n.

ar X iv : h ep - t h / 01 05 31 8 v 1 3 1 M ay 2 00 1 Trace anomaly for 4 D higher derivative scalar - dilaton theory

Trace anomaly for conformally invariant higher derivative 4D scalar-dilaton theory is obtained by means of calculating divergent part of one-loop effective action for such system. Its applications are briefly mentioned.

ar X iv : h ep - t h / 06 05 27 4 v 1 2 8 M ay 2 00 6 hep - th / 0605274 Metric and coupling reversal in string theory

Invariance under reversing the sign of the metric G M N (x) and/or the sign of the string coupling field H Universe K contains only F1 and NS5 of IIB and Heterotic SO(32).

ar X iv : m at h / 06 05 35 5 v 1 [ m at h . G R ] 1 3 M ay 2 00 6 Axes in Outer Space