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Uniformly exponential growth and mapping class groups of surfaces James W. Anderson, Javier Aramayona and Kenneth J. Shackleton
 

Summary: Uniformly exponential growth and mapping class groups of surfaces
James W. Anderson, Javier Aramayona and Kenneth J. Shackleton
27 April 2006
Abstract
We show that the mapping class group (as well as closely related groups) of an orientable
surface with finitely generated fundamental group has uniformly exponential growth. We further
demonstrate the uniformly non-amenability of many of these groups.
MSC 20F65 (primary), 20F38 (secondary)
1 Uniformly exponential growth
The main purpose of this note is to demonstrate that the mapping class group of an orientable
surface with finitely generated fundamental group has uniformly exponential growth. This result
is new for surfaces of genus at least one, with the exception of the closed surface of genus two, and
can be viewed as removing a possible avenue for showing that such mapping class groups are not
linear. In this sense, our work is similar in spirit to the recent work of Brendle and Hamidi-Tehrani
[4].
We go on to show that closely related groups of homotopy classes of homeomorphisms of surfaces, as
well as analogous groups of automorphisms of free groups, also have uniformly exponential growth.
We remark that, while the linearity of most surface mapping class groups is an open question,
most automorphism groups of free groups are known not to be linear. Specifically, as noted in
Brendle and Hamidi-Tehrani [4], it is known that Aut(Fn) is not linear for n 3 and Out(Fn)

  

Source: Aramayona, Javier - Department of Mathematics, National University of Ireland, Galway

 

Collections: Mathematics