The logarithms of Dehn twists

By introducing an invariant of loops on a compact oriented surface with one boundary component, we give an explicit formula for the action of Dehn twists on the completed group ring of the fundamental group of the surface. This invariant can be considered as ``the logarithms" of Dehn twists. The formula generalizes the classical formula describing the action on the first homology of the surface, and Morita's explicit computations of the extended first and the second Johnson homomorphisms. For the proof we use a homological interpretation of the Goldman Lie algebra in the framework of Kontsevich's formal symplectic geometry. As an application, we prove the action of the Dehn twist of a simple closed curve on the $k$-th nilpotent quotient of the fundamental group of the surface depends only on the conjugacy class of the curve in the $k$-th quotient.

通过在具有一个边界分支的紧致定向曲面上引入环的一个不变量，我们给出了德恩扭转在该曲面基本群的完备群环上作用的一个显式公式。这个不变量可被视为德恩扭转的“对数”。该公式推广了描述在曲面的一阶同调上作用的经典公式，以及森田对扩展的一阶和二阶约翰逊同态的显式计算。为了证明，我们在孔采维奇的形式辛几何框架下使用了戈德曼李代数的同调解释。作为一个应用，我们证明了一个简单闭曲线的德恩扭转在曲面基本群的第\(k\)个幂零商上的作用仅取决于该曲线在第\(k\)个商中的共轭类。