memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: 57b0e116ab371ed24a2b557766df9736a2ba0809
Parent: 20b339fbb44ee00e3ae06daa588ae05e5e32df72
Author: Pablo <pablo-pie@riseup.net>
Date: Tue, 25 Jun 2024 15:40:21 +0000

Improved a remark

Add comments on the proofs of linearity of Mod(Σ_2) by Korkmaz and Bigelow-Budney

Diffstat

2 files changed, 32 insertions, 9 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	references.bib	23	23	0
Modified	sections/introduction.tex	18	9	9

diff --git a/references.bib b/references.bib
@@ -188,6 +188,29 @@
   pages = {3097--3132}
 }
 
+@article{bigelow-budney,
+  author  = {Bigelow,  Stephen and Budney,  Ryan},
+  doi     = {10.2140/agt.2001.1.699},
+  issn    = {1472-2747},
+  journal = {Algebraic \& Geometric Topology},
+  month   = nov,
+  number  = 2,
+  pages   = {699--708},
+  title   = {The mapping class group of a genus two surface is linear},
+  volume  = {1},
+  year    = {2001},
+}
+
+@article{korkmaz-linearity,
+  author  = {Mustafa Korkmaz},
+  journal = {Turkish Journal of Mathematics},
+  pages   = {367--371},
+  title   = {On the Linearity of Certain Mapping Class Groups},
+  url     = {https://journals.tubitak.gov.tr/math/vol24/iss4/5/},
+  volume  = {24},
+  year    = {2000},
+}
+
 @book{kerekjarto,
   author    = {Kerékjártó,  Béla},
   doi       = {10.1007/978-3-642-50825-7},

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -10,7 +10,7 @@ mid-nineteenth century and is often attributed to Möbius, but at that time it
 was not yet known that all closed surfaces admit a triangulation. Radò
 \cite{rado} would go on to establish this fact in 1925.}, Kerékjártó and others
 \cite{rado, kerekjarto}. We refer the reader to \cite{thomassen} for a complete
-proof. 
+proof.
 
 \begin{theorem}[Classification of surfaces]\label{thm:classification-of-surfaces}
   Any closed connected orientable surface is homeomorphic to the connected sum
@@ -372,18 +372,18 @@ examples of mapping class groups, namely that of the torus \(\mathbb{T}^2 =
   \operatorname{SL}_2(\mathbb{Z})\).
 \end{example}
 
-% TODO: Add comments on the proof of linearity of Mod(S_2) by Korkmaz and
-% Bigelow-Budney?
 \begin{remark}
   Despite the fact \(\psi : \Mod(\mathbb{T}^2) \to
   \operatorname{SL}_2(\mathbb{Z})\) is an isomorphism, the symplectic
   representation is \emph{not} injective for surfaces of genus \(g \ge 2\) --
-  see \cite[Section~6.5]{farb-margalit} for a description of its kernel. In
-  fact, the question of existence of injective linear representations of
-  \(\Mod(\Sigma_g)\) remains wide-open. Recently, Korkmaz
-  \cite[Theorem~3]{korkmaz} established the lower bound of \(3 g - 3\) for the
-  dimension of an injective representation of \(\Mod(\Sigma_g)\) in the \(g \ge
-  3\) case -- if one such representation exists.
+  see \cite[Section~6.5]{farb-margalit} for a description of its kernel.
+  Korkmaz and Bigelow-Budney \cite{korkmaz-linearity, bigelow-budney} showed
+  there exists injective linear representations of \(\Mod(\Sigma_2)\), but the
+  question of linearity of \(\Mod(\Sigma_g)\) remains wide-open for \(g \ge
+  3\). Recently, Korkmaz \cite[Theorem~3]{korkmaz} established the lower bound
+  of \(3 g - 3\) for the dimension of an injective representation of
+  \(\Mod(\Sigma_g)\) in the \(g \ge 3\) case -- if one such representation
+  exists.
 \end{remark}
 
 Another fundamental class of examples of representations are the so called