Asymptotic Linearity of the Mapping Class Group and a Homological Version of the Nielsen–thurston Classification
نویسندگان
چکیده
We study the action of the mapping class group on the integral homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite–dimensional representation of the mapping class group. We show that this representation detects the Nielsen–Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.
منابع مشابه
The Nielsen-thurston Classification of Mapping Classes Is Determined by Tqft
For each fixed n ≥ 2 we show how the Nielsen-Thurston classification of mapping classes for a closed surface of genus g ≥ 2 is determined by the sequence of quantum SU(n)-representations (ρk)k∈N. That this is the case is a consequence of the asymptotic faithfulness property proved in [A3]. We here provide explicit conditions on (ρk(φ))k∈N, which determines the NielsenThurston type of any mappin...
متن کاملAn Effective Algebraic Detection of the Nielsen–thurston Classification of Mapping Classes
In this article, we propose two algorithms for determining the Nielsen-Thurston classification of a mapping class ψ on a surface S. We start with a finite generating set X for the mapping class group and a word ψ in 〈X〉. We show that if ψ represents a reducible mapping class in Mod(S) then ψ admits a canonical reduction system whose total length is exponential in the word length of ψ. We use th...
متن کاملCharacterization of weak fixed point property for new class of set-valued nonexpansive mappings
In this paper, we introduce a new class of set-valued mappings which is called MD-type mappings. This class of mappings is a set-valued case of a class of the D-type mappings. The class of D-type mappings is a generalization of nonexpansive mappings that recently introduced by Kaewkhao and Sokhuma. The class of MD-type mappings includes upper semi-continuous Suzuki type mappings, upper semi-con...
متن کاملClassification of Weil-petersson Isometries
This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmüller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as WeilPetersson isometries which is parallel to the Thurston classification. The second result concerns the asymptotic behavior of t...
متن کاملThe geometry of the curve graph of a right-angled Artin group
We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we are able to develop a Nielsen–Thurston classification for elements in the rig...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009