  The $\beta$ decay of the neutron-rich $^{134}$In and $^{135}$In was
investigated experimentally in order to provide new insights into the nuclear
structure of the tin isotopes with magic proton number $Z=50$ above the $N=82$
shell. The $\beta$-delayed $\gamma$-ray spectroscopy measurement was performed
at the ISOLDE facility at CERN, where indium isotopes were selectively
laser-ionized and on-line mass separated. Three $\beta$-decay branches of
$^{134}$In were established, two of which were observed for the first time.
Population of neutron-unbound states decaying via $\gamma$ rays was identified
in the two daughter nuclei of $^{134}$In, $^{134}$Sn and $^{133}$Sn, at
excitation energies exceeding the neutron separation energy by 1 MeV. The
$\beta$-delayed one- and two-neutron emission branching ratios of $^{134}$In
were determined and compared with theoretical calculations. The $\beta$-delayed
one-neutron decay was observed to be dominant $\beta$-decay branch of
$^{134}$In even though the Gamow-Teller resonance is located substantially
above the two-neutron separation energy of $^{134}$Sn. Transitions following
the $\beta$ decay of $^{135}$In are reported for the first time, including
$\gamma$ rays tentatively attributed to $^{135}$Sn. In total, six new levels
were identified in $^{134}$Sn on the basis of the $\beta \gamma \gamma$
coincidences observed in the $^{134}$In and $^{135}$In $\beta$ decays. A
transition that might be a candidate for deexciting the missing neutron
single-particle $13/2^+$ state in $^{133}$Sn was observed in both $\beta$
decays and its assignment is discussed. Experimental level schemes of
$^{134}$Sn and $^{135}$Sn are compared with shell-model predictions. Using the
fast timing technique, half-lives of the $2^+$, $4^+$ and $6^+$ levels in
$^{134}$Sn were determined.
