Why subscribe/unsubscribe button in the custom entity named group cannot appear?
Since you are taking over the display of Organic Groups nodes using your custom page callback path group/%, DrupalGap doesn't know to apply the field format view functions to your fields. Essentially you need to manually create/render your own widgets in your page_callback:

var content = {};
content.title = { /* ... */ };
content.subscribe = { /* ... */ };
return content;


There are helper functions in the og.js file related to joining/leaving groups that will be of help. From there just use simple HTML/CSS/JS to build your button(s) (perhaps with an onclick handler) to help the user interact with the group subscriptions.

If you want to call the og_ui_field_formatter_view() function directly (not recommended) you can, you just need to pass it all the correct parameters:

(entity_type, entity, field, instance, langcode, items, display)


the entity_type is typically 'node'
the entity will be the fully loaded JSON of the entity (already delivered to your page_callback)
the field and instance can be retrieved from http://api.drupalgap.org/7/global.html#drupalgap_field_info_field and http://api.drupalgap.org/7/global.html#drupalgap_field_info_instance respectively.
the langcode will most likely be und unless you are using a multi-lingual site
the items will be the value(s) within the field itself
the display you should default to { type: 'og_group_subscribe' }. 


Not the easiest things to do to be honest, and you'll probably find yourself happier in the long run by building your own custom widget to subscribe/unsubscribe from the Group.

Feel free to copy any code you need from the og.js file to make your life easier.