diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 00553edca3259..17a830788be3f 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -902,6 +902,10 @@ Groupby/resample/rolling
 - Bug in :meth:`Series.groupby` would raise ``ValueError`` when grouping by :class:`PeriodIndex` level (:issue:`34010`)
 - Bug in :meth:`GroupBy.agg`, :meth:`GroupBy.transform`, and :meth:`GroupBy.resample` where subclasses are not preserved (:issue:`28330`)
 - Bug in :meth:`GroupBy.rolling.apply` ignores args and kwargs parameters (:issue:`33433`)
+- Bug in :meth:`core.groupby.DataFrameGroupBy.apply` where the output index shape for functions returning a DataFrame which is equally indexed
+  to the input DataFrame is inconsistent. An internal heuristic to detect index mutation would behave differently for equal but not identical
+  indices. In particular, the result index shape might change if a copy of the input would be returned.
+  The behaviour now is consistent, independent of internal heuristics. (:issue:`31612`, :issue:`14927`, :issue:`13056`)
 
 Reshaping
 ^^^^^^^^^
diff --git a/pandas/_libs/reduction.pyx b/pandas/_libs/reduction.pyx
index 0988cd7ff0dde..18422c2f86129 100644
--- a/pandas/_libs/reduction.pyx
+++ b/pandas/_libs/reduction.pyx
@@ -502,7 +502,7 @@ def apply_frame_axis0(object frame, object f, object names,
             # Need to infer if low level index slider will cause segfaults
             require_slow_apply = i == 0 and piece is chunk
             try:
-                if piece.index is not chunk.index:
+                if not piece.index.equals(chunk.index):
                     mutated = True
             except AttributeError:
                 # `piece` might not have an index, could be e.g. an int
diff --git a/pandas/tests/groupby/test_apply.py b/pandas/tests/groupby/test_apply.py
index e2b5118922a5a..bc8067212d60e 100644
--- a/pandas/tests/groupby/test_apply.py
+++ b/pandas/tests/groupby/test_apply.py
@@ -190,6 +190,46 @@ def f_constant_df(group):
         assert names == group_names
 
 
+def test_apply_fast_slow_identical():
+    # GH 31613
+
+    df = DataFrame({"A": [0, 0, 1], "b": range(3)})
+
+    # For simple index structures we check for fast/slow apply using
+    # an identity check on in/output
+    def slow(group):
+        return group
+
+    def fast(group):
+        return group.copy()
+
+    fast_df = df.groupby("A").apply(fast)
+    slow_df = df.groupby("A").apply(slow)
+
+    tm.assert_frame_equal(fast_df, slow_df)
+
+
+@pytest.mark.parametrize(
+    "func",
+    [
+        lambda x: x,
+        lambda x: x[:],
+        lambda x: x.copy(deep=False),
+        lambda x: x.copy(deep=True),
+    ],
+)
+def test_groupby_apply_identity_maybecopy_index_identical(func):
+    # GH 14927
+    # Whether the function returns a copy of the input data or not should not
+    # have an impact on the index structure of the result since this is not
+    # transparent to the user
+
+    df = pd.DataFrame({"g": [1, 2, 2, 2], "a": [1, 2, 3, 4], "b": [5, 6, 7, 8]})
+
+    result = df.groupby("g").apply(func)
+    tm.assert_frame_equal(result, df)
+
+
 def test_apply_with_mixed_dtype():
     # GH3480, apply with mixed dtype on axis=1 breaks in 0.11
     df = DataFrame(