Trac #28078: braid groups are infinite

unless they have one strand

URL: https://trac.sagemath.org/28078
Reported by: chapoton
Ticket author(s): Travis Scrimshaw
Reviewer(s): Travis Scrimshaw

src/sage/groups/braid.py

@@ -62,26 +62,28 @@
 #
 #  The full text of the GPL is available at:
 #
-#                  http://www.gnu.org/licenses/
+#                  https://www.gnu.org/licenses/
 ##############################################################################
 
 import six
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import IntegerRing
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.misc.cachefunc import cached_method
+from sage.categories.groups import Groups
 from sage.groups.free_group import FreeGroup, is_FreeGroup
 from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
 from sage.matrix.constructor import identity_matrix, matrix
 from sage.combinat.permutation import Permutations
 from sage.categories.action import Action
 from sage.sets.set import Set
-from sage.groups.finitely_presented import FinitelyPresentedGroup, FinitelyPresentedGroupElement
+from sage.groups.finitely_presented import FinitelyPresentedGroup
 from sage.groups.artin import FiniteTypeArtinGroup, FiniteTypeArtinGroupElement
 from sage.misc.package import PackageNotFoundError
 from sage.structure.richcmp import richcmp, rich_to_bool
 from sage.misc.superseded import deprecated_function_alias
 
+
 class Braid(FiniteTypeArtinGroupElement):
     """
     An element of a braid group.
@@ -1019,12 +1021,11 @@ def _left_normal_form_coxeter(self):
             Remove this method and use the default one from
             :meth:`sage.groups.artin.FiniteTypeArtinGroupElement.left_normal_form`.
         """
-        n = self.parent().strands()
         delta = 0
         Delta = self.parent()._coxeter_group.long_element()
         sr = self.parent()._coxeter_group.simple_reflections()
         l = self.Tietze()
-        if l == ():
+        if not l:
             return (0,)
         form = []
         for i in l:
@@ -1430,7 +1431,8 @@ def __init__(self, names):
             sage: B1
             Braid group on 5 strands
             sage: TestSuite(B1).run()
-
+            sage: B1.category()
+            Category of infinite groups
 
         Check that :trac:`14081` is fixed::
 
@@ -1441,10 +1443,10 @@ def __init__(self, names):
 
         Check that :trac:`15505` is fixed::
 
-            sage: B=BraidGroup(4)
+            sage: B = BraidGroup(4)
             sage: B.relations()
             (s0*s1*s0*s1^-1*s0^-1*s1^-1, s0*s2*s0^-1*s2^-1, s1*s2*s1*s2^-1*s1^-1*s2^-1)
-            sage: B=BraidGroup('a,b,c,d,e,f')
+            sage: B = BraidGroup('a,b,c,d,e,f')
             sage: B.relations()
             (a*b*a*b^-1*a^-1*b^-1,
              a*c*a^-1*c^-1,
@@ -1473,7 +1475,12 @@ def __init__(self, names):
             rels.append(free_group([i, i+1, i, -i-1, -i, -i-1]))
             for j in range(i+2, n+1):
                 rels.append(free_group([i, j, -i, -j]))
-        FinitelyPresentedGroup.__init__(self, free_group, tuple(rels))
+        if n:
+            cat = Groups().Infinite()
+        else:
+            cat = Groups().Finite()
+        FinitelyPresentedGroup.__init__(self, free_group, tuple(rels),
+                                        category=cat)
         self._nstrands = n+1
         self._coxeter_group = Permutations(self._nstrands)
 
@@ -1583,7 +1590,7 @@ def an_element(self):
 
         EXAMPLES::
 
-            sage: B=BraidGroup(2)
+            sage: B = BraidGroup(2)
             sage: B.an_element()
             s
         """
@@ -1675,7 +1682,7 @@ def _LKB_matrix_(self, braid, variab):
 
         TESTS::
 
-            sage: B=BraidGroup(3)
+            sage: B = BraidGroup(3)
             sage: B._LKB_matrix_((2, 1, 2), 'x, y')
             [             0 -x^4*y + x^3*y         -x^4*y]
             [             0         -x^3*y              0]
@@ -2199,7 +2206,7 @@ def BraidGroup(n=None, names='s'):
     or the number of generators and the prefix of the names to be
     given. The default prefix is ``'s'`` ::
 
-        sage: B=BraidGroup(3); B.generators()
+        sage: B = BraidGroup(3); B.generators()
         (s0, s1)
         sage: BraidGroup(3, 'g').generators()
         (g0, g1)

src/sage/groups/finitely_presented.py

@@ -742,6 +742,7 @@ def make_confluent(self):
         except ValueError:
             raise ValueError('could not make the system confluent')
 
+
 class FinitelyPresentedGroup(GroupMixinLibGAP, UniqueRepresentation,
     Group, ParentLibGAP):
     """
@@ -783,7 +784,7 @@ class FinitelyPresentedGroup(GroupMixinLibGAP, UniqueRepresentation,
     """
     Element = FinitelyPresentedGroupElement
 
-    def __init__(self, free_group, relations):
+    def __init__(self, free_group, relations, category=None):
         """
         The Python constructor.
 
@@ -808,9 +809,9 @@ def __init__(self, free_group, relations):
         self._free_group = free_group
         self._relations = relations
         self._assign_names(free_group.variable_names())
-        parent_gap = free_group.gap() / libgap([ rel.gap() for rel in relations])
+        parent_gap = free_group.gap() / libgap([rel.gap() for rel in relations])
         ParentLibGAP.__init__(self, parent_gap)
-        Group.__init__(self)
+        Group.__init__(self, category=category)
 
     def _repr_(self):
         """