Event Espresso 4.2 and 4.3 capability filter examples. Thread: http://eventespresso.com/topic/ee4-creating-a-role-to-just-manage-espresso-events/

functions.php

<?php
//* Please do NOT include the opening php tag, except of course if you're starting with a blank file

// give user roles that can 'publish_posts' access to events, registrations, transactions, registration forms
// roles will also need 'manage_options' cap

add_filter ('FHEE_management_capability', 'my_custom_menu_management_capability');

function my_custom_menu_management_capability() {
    return 'publish_posts';
}

add_filter ('FHEE_espresso_events_capability', 'my_custom_espresso_events_capability');

function my_custom_espresso_events_capability($capability){
    return 'publish_posts';
}

add_filter ('FHEE_espresso_registrations_capability', 'my_custom_espresso_registrations_capability');

function my_custom_espresso_registrations_capability($capability){
    return 'publish_posts';
}

add_filter ('FHEE_espresso_transactions_capability', 'my_custom_espresso_transactions_capability');

function my_custom_espresso_transactions_capability($capability){
    return 'publish_posts';
}

add_filter ('FHEE_espresso_registration_form_capability', 'my_custom_espresso_transactions_capability');

function my_custom_espresso_registration_form_capability($capability){
    return 'publish_posts';
}