Characterizing SL2S galaxy groups using the Einstein radius
  We analyzed the Einstein radius, $\theta_E$, in our sample of SL2S galaxy
groups, and compared it with $R_A$ (the distance from the arcs to the center of
the lens), using three different approaches: 1.- the velocity dispersion
obtained from weak lensing assuming a Singular Isothermal Sphere profile
($\theta_{E,I}$), 2.- a strong lensing analytical method ($\theta_{E,II}$)
combined with a velocity dispersion-concentration relation derived from
numerical simulations designed to mimic our group sample, 3.- strong lensing
modeling ($\theta_{E,III}$) of eleven groups (with four new models presented in
this work) using HST and CFHT images. Finally, $R_A$ was analyzed as a function
of redshift $z$ to investigate possible correlations with L, N, and the
richness-to-luminosity ratio (N/L). We found a correlation between $\theta_{E}$
and $R_A$, but with large scatter. We estimate $\theta_{E,I}$ = (2.2 $\pm$ 0.9)
+ (0.7 $\pm$ 0.2)$R_A$, $\theta_{E,II}$ = (0.4 $\pm$ 1.5) + (1.1 $\pm$
0.4)$R_A$, and $\theta_{E,III}$ = (0.4 $\pm$ 1.5) + (0.9 $\pm$ 0.3)$R_A$ for
each method respectively. We found a weak evidence of anti-correlation between
$R_A$ and $z$, with Log$R_A$ = (0.58$\pm$0.06) - (0.04$\pm$0.1)$z$, suggesting
a possible evolution of the Einstein radius with $z$, as reported previously by
other authors. Our results also show that $R_A$ is correlated with L and N
(more luminous and richer groups have greater $R_A$), and a possible
correlation between $R_A$ and the N/L ratio. Our analysis indicates that $R_A$
is correlated with $\theta_E$ in our sample, making $R_A$ useful to
characterize properties like L and N (and possible N/L) in galaxy groups.
Additionally, we present evidence suggesting that the Einstein radius evolves
with $z$.
