Dica testada no Debian Stretch.

Telegram Desktop não abre e apresenta um erro assim:

(telegram-desktop:6057): GLib-GObject-WARNING **: cannot register existing type 'GdkDisplayManager'
(telegram-desktop:6057): GLib-CRITICAL **: g_once_init_leave: assertion 'result != 0' failed
(telegram-desktop:6057): GLib-GObject-CRITICAL **: g_object_new: assertion 'G_TYPE_IS_OBJECT (object_type)' failed
(telegram-desktop:6057): GLib-GObject-WARNING **: invalid (NULL) pointer instance
(telegram-desktop:6057): GLib-GObject-CRITICAL **: g_signal_connect_data: assertion 'G_TYPE_CHECK_INSTANCE (instance)' failed
(telegram-desktop:6057): GLib-GObject-WARNING **: invalid (NULL) pointer instance
(telegram-desktop:6057): GLib-GObject-CRITICAL **: g_signal_connect_data: assertion 'G_TYPE_CHECK_INSTANCE (instance)' failed
(telegram-desktop:6057): GLib-GObject-WARNING **: cannot register existing type 'GdkDisplay'
(telegram-desktop:6057): GLib-CRITICAL **: g_once_init_leave: assertion 'result != 0' failed
(telegram-desktop:6057): GLib-GObject-CRITICAL **: g_type_register_static: assertion 'parent_type > 0' failed
(telegram-desktop:6057): GLib-CRITICAL **: g_once_init_leave: assertion 'result != 0' failed
(telegram-desktop:6057): GLib-GObject-CRITICAL **: g_object_new: assertion 'G_TYPE_IS_OBJECT (object_type)' failed
Falha de segmentação

Solução

Instalar o "libappindicator3-1".

sudo apt install libappindicator3-1

Após isso o programa irá funcionar.

Fonte: #859172 - telegram-desktop: Segmentfault due to lack of libappindicator3-1 library dep - Debian Bug report logs