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You need get_categories() with an include argument:

if (isset($_GET['cat'])) {
  $this_category = get_categories(
    array(
      'include' => $_GET['cat']
    )
  );
  var_dump($this_category);
}

You will get an array of stdClass objects containing category data. There are several ways to parse that but this might do:

if (isset($_GET['cat'])) {
  $this_category = get_categories(
    array(
      'include' => $_GET['cat']
    )
  );
  //   var_dump($this_category);
  $cats = wp_list_pluck($this_category,'slug');
  var_dump($cats);
  $needle = array('groups-london','groups-cambridge');
  var_dump(array_intersect($needle,$cats));
  if(array_intersect($needle,$cats)){
    include(TEMPLATEPATH.'/groups.php');
  }else{
    include(TEMPLATEPATH.'/general.php');
  }
}