|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'500'096
Articles rated: 2609
18 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Pappus theorem, schwartz representations and anosov representations | | Thierry Barbot
; Gye-Seon Lee
; Viviane Pardini Valério
; | | Date: |
13 Oct 2016 | | Abstract: | In the paper Pappus’s theorem and the modular group [13], R. Schwartz
constructed a 2-dimensional family of faithful representations $
ho$$Theta$
of the modular group PSL(2, Z) into the group G of projective symmetries of the
projective plane via Pappus Theorem. If PSL(2, Z)o denotes the unique index 2
subgroup of PSL(2, Z) and PGL(3, R) the subgroup of G consisting of projective
transformations, then the image of PSL(2, Z)o under $
ho$$Theta$ is in PGL(3,
R). The representations $
ho$$Theta$ share a very interesting property with
Anosov representations of surface groups into PGL(3, R): It preserves a
topological circle in the flag variety. However, the representation
$
ho$$Theta$ itself cannot be Anosov since the Gromov boundary of PSL(2, Z)
is a Cantor set and not a circle. In her PhD Thesis [15], V. P. Val{’e}rio
elucidated the Anosov-like feature of the Schwartz representations by showing
that for each representation $
ho$$Theta$, there exists an 1-dimensional
family of representations ($
ho$ $epsilon$ $Theta$) $epsilon$$in$R of
PSL(2, Z)o into PGL(3, R) such that $
ho$ 0 $Theta$ is the restriction of the
Schwartz representation $
ho$$Theta$ to PSL(2, Z)o and $
ho$ $epsilon$
$Theta$ is Anosov for every $epsilon$ extless{} 0. This result was
announced and presented in her paper [14]. In the present paper, we extend and
improve her work. For every representation $
ho$$Theta$, we build a
2-dimensional family of representations ($
ho$ $lambda$ $Theta$)
$lambda$$in$R 2 of PSL(2, Z)o into PGL(3, R) such that $
ho$ $lambda$
$Theta$ = $
ho$ $epsilon$ $Theta$ for $lambda$ = ($epsilon$, 0) and
$
ho$ $lambda$ $Theta$ is Anosov for every $lambda$ $in$ R $ullet$ ,
where R $ullet$ is an open set of R 2 containing {($epsilon$, 0) |
$epsilon$ extless{} 0}. Moreover, among the 2-dimensional family of new
Anosov representations, an 1-dimensional subfamily of representations can
extend to representations of PSL(2, Z) into G, and therefore the Schwartz
representations are, in a sense, on the boundary of the Anosov representations
in the space of all representations of PSL(2, Z) into G. | | Source: | arXiv, 1610.4049 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER