Toroidal Dehn Fillings on Hyperbolic 3-manifolds

Introduction Preliminary lemmas $\hat \Gamma_a^+$ has no interior vertex Possible components of $\hat \Gamma_a^+$ The case $n_1, n_2 > 4$ Kleinian graphs If $n_a=4$, $n_b \geq 4$ and $\hat \Gamma_a^+$ has a small component then $\Gamma_a$ is kleinian If $n_a=4$, $n_b \geq 4$ and $\Gamma_b$is non-positive then $\hat \Gamma_a^+$ has no small component If $\Gamma_b$ is non-positive and $n_a=4$ then $n_b \leq 4$ The case $n_1 = n_2 = 4$ and $\Gamma_1, \Gamma_2$ non-positive The case $n_a = 4$, and $\Gamma_b$ positive The case $n_a=2$, $n_b \geq 3$, and $\Gamma_b$ positive The case $n_a = 2$, $n_b > 4$, $\Gamma_1, \Gamma_2$ non-positive, and $\text{max}(w_1 + w_2, \, \, w_3 + w_4) = 2n_b-2$ The case $n_a = 2$, $n_b > 4$, $\Gamma_1, \Gamma_2$ non-positive, and $w_1 = w_2 = n_b$ $\Gamma_a$ with $n_a \leq 2$ The case $n_a = 2$, $n_b=3$ or $4$, and $\Gamma_1, \Gamma_2$ non-positive Equidistance classes The case $n_b = 1$ and $n_a = 2$ The case $n_1 = n_2 = 2$ and $\Gamma_b$ positive The case $n_1 = n_2 = 2$ and both $\Gamma_1, \Gamma_2$ non-positive The main theorems The construction of $M_i$ as a double branched cover The manifolds $M_i$ are hyperbolic Toroidal surgery on knots in $S^3$ Bibliography.