GEOMETRY OF THE OUTER AUTOMORPHISM GROUP OF A FREE GROUP

Project Details

Description

The Principal Investigator (jointly with Michael Handel of Lehman College,CUNY) is studying asymptotic and algebraic properties of Out(F_n) and X_n, the outer automorphism group and outer space of a free group of rank n. The method of study is to develop an analogue of the hierarchy theory of Masur and Minsky used to study MCG(S), the mapping class group of a surface S. Based on his recent discoveries with Handel of distortion and nondistortion of various natural subgroups of Out(F_n), the PI is currently focussed on using free splittings of the free group F_n as a natural analogue of essential curve systems on surfaces, and on the 'refinement complex' of a free splitting as a natural analogue (at least in certain cases) for the curve complex of a subsurface. In particular, the PI is studying how to use projection maps defined on the spine of X_n, whose targets are refinement complexes and closely related objects. Using these projection maps, the central focus of the PI's research at the moment is to formulate an appropriate analogue for Out(F_n) of the Masur--Minsky sum, a very useful formula for the word metric in MCG(S). Potential mathematical applications include an improved understanding of the computational complexity of the conjugacy problem for Out(F_n). The PI also continues to work on a longer term joint project with Handel on subgroup classification theory for Out(F_n). In a separate project (joint with Jason Behrstock of Lehman College, CUNY) the PI works on applying hierarchy theory for MCG(S) to obtain a precise understanding of the computational complexity of the conjugacy problem for MCG(S).Geometric group theory is the study of symmetry groups of geometric objects. In some cases the geometric object was understood long before its symmetry group, for instance the flat plane was known in antiquity, but its symmetries --- its rigid motions --- were fully understood only in the 19th century. In other cases, the abstract group was known before the discovery of a geometry with that symmetry group. The Principal Investigator studies an abstract group called the 'outer automorphism group of a free group of rank n', denoted Out(F_n), which was discovered in the first half of the 20th century. Its associated geometric object, the 'outer space of rank n' denoted X_n, was discovered only in the 1980's by the mathematicians Marc Culler and Karen Vogtmann. The space X_n, a key object for understanding finite networks, may be regarded as a single package encompassing a complicated array of networks known as 'marked graphs' that can be obtained by distorting one very simple network, the 'mathematician's rose', a collection of n loops conjoined at a single point. Following a broad research program for geometric group theory that was laid out in the late 1970's by the mathematician Michael Gromov, the PI's study of Out(F_n) is closely intertwined with a study of the asymptotic behavior of X_n, meaning its behaviour very very far away from a fixed point of view. The PI's asymptotic analysis of Out(F_n) and X_n involves comparing efficient versus inefficient distortions of marked graphs. The central focus of the PI's research at the moment is to discover a general formula for measuring efficient distortion of marked graphs.
StatusFinished
Effective start/end date6/1/105/31/12

Funding

  • National Science Foundation (National Science Foundation (NSF))

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