def __init__(self, data, group=None):
"""
Builds a FreeGroupAutomorphism from data.
INPUT:
- ``data`` - the data used to build the morphism
- ``group`` - an optional free group
"""
if group is not None and not isinstance(group, FreeGroup):
raise ValueError, "the group must be a Free Group"
WordMorphism.__init__(self, data)
if group is None:
A = AlphabetWithInverses(self.domain().alphabet())
F = FreeGroup(A)
else:
F = group
A = group.alphabet()
self._domain = F
self._codomain = F # unuseful... consistency with WordMorphism
for letter in self._morph.keys():
self._morph[letter] = F.reduce(self._morph[letter])
self._morph[A.inverse_letter(letter)] = F.inverse_word(
self._morph[letter])
def __init__(self,data,group=None):
"""
Builds a FreeGroupAutomorphism from data.
INPUT:
- ``data`` - the data used to build the morphism
- ``group`` - an optional free group
"""
if group is not None and not isinstance(group, FreeGroup):
raise ValueError, "the group must be a Free Group"
WordMorphism.__init__(self,data)
if group is None:
A = AlphabetWithInverses(self.domain().alphabet())
F = FreeGroup(A)
else:
F = group
A = group.alphabet()
self._domain = F
self._codomain = F # unuseful... consistency with WordMorphism
for letter in self._morph.keys():
self._morph[letter]=F.reduce(self._morph[letter])
self._morph[A.inverse_letter(letter)] = F.inverse_word(self._morph[letter])
def Hilion_parabolic(k=1):
"""
Automorphism given in Section 2 of [Hilion].
This automorphism has a parabolic orbit inside F_4.
REFERENCES:
[Hilion] A. Hilion
"""
F=FreeGroup(4)
phi=FreeGroupAutomorphism("a->a,b->ba,c->caa,d->dc",F)
if k>1:
phi=phi*pow(F.dehn_twist(c,a),k-1)
return phi
def Hilion_parabolic(k=1):
"""
Automorphism given in Section 2 of [Hilion].
This automorphism has a parabolic orbit inside F_4.
REFERENCES:
[Hilion] A. Hilion, Dynamique des automorphismes des groupes
libres, Thesis (Toulouse, 2004).
"""
F=FreeGroup(4)
phi=FreeGroupAutomorphism("a->a,b->ba,c->caa,d->dc",F)
if k>1:
phi=phi*pow(F.dehn_twist(c,a),k-1)
return phi
def Hilion_parabolic(k=1):
"""
Automorphism given in Section 2 of [Hilion].
This automorphism has a parabolic orbit inside F_4.
REFERENCES:
[Hilion] A. Hilion, Dynamique des automorphismes des groupes
libres, Thesis (Toulouse, 2004).
"""
F = FreeGroup(4)
phi = FreeGroupAutomorphism("a->a,b->ba,c->caa,d->dc", F)
if k > 1:
phi = phi * pow(F.dehn_twist(c, a), k - 1)
return phi
def Bestvina_Handel_train_track_1_9():
"""
Automorphism given as Example 1.9 in [BH-train-track]
This automorphism cannot be represented by an absolute train-track. But
the representation on the rose is a relative train-track.
REFERENCES:
[BH-train-track] M. Bestvina, M. Handel, Train tracks and
automorphisms of free groups, Annals of Math, 135, 1-51, 1992.
"""
return FreeGroupAutomorphism("a->ba,b->bba,c->cAbaB", FreeGroup(3))
def Bestvina_Handel_train_track_3_6():
"""
Automorphism given as Example 3.6 in [BH-train-track].
This automorphism is train-track on the rose and has an indivisble
Nielsen path in A.b which is essential.
REFERENCES:
[BH-train-track] M. Bestvina, M. Handel, Train tracks and
automorphisms of free groups, Annals of Math, 135, 1-51, 1992.
"""
return FreeGroupAutomorphism("a->ba,b->bba", FreeGroup(2))
def Levitt_Lustig_periodic():
"""
Automorphism of F_3 given in Section 2 of [LL-periodic].
This is an atoroidal iwip. It is positive and thus train-track
on the rose.
REFERENCES:
[LL-periodic] G. Levitt, and M. Lustig, Automorphisms of free
groups have asymptotically periodic dynamics,
"""
return FreeGroupAutomorphism("a->cb,b->a,c->ba", FreeGroup(3))
def Clay_Pettet_twisting_out():
"""
Automorphism of F_3 given in Section 2 of [CP-twisting].
This is an atoroidal iwip. It is positive and thus train-track
on the rose.
REFERENCES:
[CP-twisting] M. Clay, and A. Pettet, Twisting out fully
irreducible automorphisms, ArXiv:0906.4050
"""
return FreeGroupAutomorphism("a->b,b->c,c->ab", FreeGroup(3))
def Bestvina_Handel_train_track_1_1():
"""
Automorphism given as Example 1.1 in [BH-train-track].
This automorphism is iwip and not geometric nor
parageometric. Its representative on the rose is
train-track. Its inverse is also train-track on the rose.
REFERENCES:
[BH-train-track] M. Bestvina, M. Handel, Train tracks and
automorphisms of free groups, Annals of Math, 135, 1-51, 1992.
"""
return FreeGroupAutomorphism("a->b,b->c,c->d,d->ADBC", FreeGroup(4))
def Bestvina_Handel_train_track_5_16():
"""
Automorphism given as Example 5.16 in [BH-train-track].
This automorphism occurs as a pseudo-Anosov homeomorphism of
the four-times punctured phere. Thus is it reducible.
REFERENCES:
[BH-train-track] M. Bestvina, M. Handel, Train tracks and
automorphisms of free groups, Annals of Math, 135, 1-51, 1992.
"""
return FreeGroupAutomorphism("a->a,b->CAbac,c->CAbacacACABac",
FreeGroup(3))
def Cohen_Lustig_7_3():
"""
Automorphism given as example 7.3 in [CL-dynamics].
This is an atoroidal iwip.
REFERENCES:
[CL-dynamics] M. Cohen, M. Lustig, on the dynamics and the
fixed subgroup of a free group automorphism, Inventiones
Math. 96, 613-638, 1989.
"""
return FreeGroupAutomorphism("a->cabaa,b->baa,c->caba", FreeGroup(3))
def Cohen_Lustig_1_6():
"""
Automorphism given as example 1.6 in [CL-dynamics].
It is reducible.
REFERENCES:
[CL-dynamics] M. Cohen, M. Lustig, on the dynamics and the
fixed subgroup of a free group automorphism, Inventiones
Math. 96, 613-638, 1989.
"""
return FreeGroupAutomorphism(
"a->cccaCCC,b->CaccAbC,c->accAbccaCCBaCCAccccACCC", FreeGroup(3))
def Handel_Mosher_axes_5_5():
"""
Automorphism given in Section 5.5 of [HM-axes]
This automorphism phi is iwip and not geometric. Both phi and
phi.inverse() are train-track on the rose. phi has expansion
factor 6.0329 while phi.inverse() has expansion factor
4.49086.
REFERENCES:
[HM-axes] M. Handel, L. Mosher, axes
in Outer space, Mem. Amer. Math. Soc. 213, 2011.
"""
return FreeGroupAutomorphism("a->bacaaca,b->baca,c->caaca",
FreeGroup(3))
def Handel_Mosher_parageometric_1():
"""
Automorphism given in the introduction of [HM-parageometric].
This automorphism phi is iwip, not geometric and
parageometric. Both phi and phi.inverse() are train-track on
the rose. phi has expansion factor 1.46557 while phi.inverse()
has expansion factor 1.3247.
REFERENCES:
[HM-parageometric] M. Handel, L. Mosher, parageometric outer
automorphisms of free groups, Transactions of
Amer. Math. Soc. 359, 3153-3183, 2007.
"""
return FreeGroupAutomorphism("a->ac,b->a,c->b", FreeGroup(3))
def Handel_Mosher_axes_3_4():
"""
Automorphism given in Section 3.4 of [HM-axes]
This automorphism is iwip, not geometric and is train-track on
the rose. It has expansion factor 4.0795. Its inverse is not
train-track on the rose and has expansion factor 2.46557. It
also appears in Section 5.5 of the paper.
REFERENCES:
[HM-axes] M. Handel, L. Mosher, axes
in Outer space, Mem. Amer. Math. Soc. 213, 2011.
"""
A = AlphabetWithInverses(['a', 'g', 'f'], ['A', 'G', 'F'])
return FreeGroupAutomorphism("a->afgfgf,f->fgf,g->gfafg", FreeGroup(A))
def Handel_Mosher_inverse_with_same_lambda():
"""
Example given in the introduction of [HM-parageometric].
This is an iwip automorphisms that has the same expansion factor as its
inverse: 3.199. It is not geometric and not parageometric.
REFERECENCES:
[HM-parageometric] M. Handel, L. Mosher, parageometric outer
automorphisms of free groups, Transactions of
Amer. Math. Soc. 359, 3153-3183, 2007.
"""
F = FreeGroup(3)
theta = pow(FreeGroupAutomorphism("a->b,b->c,c->Ba", F), 4)
psi = FreeGroupAutomorphism("a->b,b->a,c->c", F)
return psi * theta * psi * theta.inverse()
def Turner_Stallings():
"""
Automorphism of F_4 given in [Turner].
This automorphism comes from an idea of Stallings and although
it is very short, it has a very long fixed word.
It is a reducible automorphism.
REFERENCES:
[Turner] E. C. Turner, Finding indivisible Nielsen paths for a
train tracks map, Proc. of a work- shop held at Heriot-Watt Univ.,
Edinburg, 1993 (Lond. Math. Soc. Lect. Note Ser., 204), Cambridge,
Cambridge Univ. Press., 1995, 300-313.
"""
return FreeGroupAutomorphism("a->dac,b->CADac,c->CABac,d->CAbc",
FreeGroup(4))
def Bestvina_Handel_surface_homeo():
"""
Automorphism of F_4 given in [BH] see also [Kapovich].
This is a pseudo-Anosov mapping class of the 5-punctured
sphere. Thus this is not an iwip. However, its representative
on the rose in train-track.
REFERENCES:
[BH] M. Bestvina, and M. Handel, Train-tracks for surface
homeomorphisms. Topology 34 (1995), no. 1, 109-140
[Kapovich] Ilya Kapovich, Algorithmic detectability of iwip
automorphisms, arXiv:1209.3732
"""
return FreeGroupAutomorphism("a->b,b->c,c->dA,d->DC", FreeGroup(4))
def _build_endmap(self, consolidate=False):
"""
Builds the map on ends for graph_map. Called when Core is initialized.
"""
alph = self._graph_map.domain()._alphabet
inv_alph = self._inv_graph_map.domain()._alphabet
FA = FreeGroup(alph)
FB = FreeGroup(inv_alph)
for x in self._graph_map.domain().edge_labels():
X = alph.inverse_letter(x)
self._signed_ends[x] = []
self._signed_ends[X] = []
for x in self._inv_graph_map.domain().edge_labels():
X = inv_alph.inverse_letter(x)
inv_image_x = self._inv_graph_map(x)
vp = '' # vanishing path
for a in inv_image_x:
A = alph.inverse_letter(a)
if vp == '' or vp[
-1] != X: # coming from + side, no cancellation
self._signed_ends[a].append(vp + x)
else: # coming from - side, cancels x
self._signed_ends[a].append('-' + vp) # - + vp #
# add to vanishing path
vp = str(FB.reduced_product(str(self._graph_map(A)), vp))
# repeat for inverse, we've added to the vanishing path because we are
# coming from the other side
if vp == '' or vp[
-1] != X: # coming from - side, no cancellation
self._signed_ends[A].append('-' + vp + x) # - + vp + x
else: # coming from + side, cancels x
self._signed_ends[A].append(vp)
if consolidate:
for a in self._signed_ends.keys():
removed_end = True
N = 2 * inv_alph.cardinality(
) # should take into account the actual vertex...
while removed_end:
removed_end = False
prefix = []
for e in self._signed_ends[a]:
prefix.append(e[:-1])
prefix_counts = dict(
(p, prefix.count(p)) for p in set(prefix))
for p in prefix_counts.keys():
p_opp = p[1:] if p[:1] == '-' else '-' + p
removed_ends = []
tails = []
if prefix_counts[p] == N - 1:
removed_end = True
for e in self._signed_ends[a]:
if e[:-1] == p:
removed_ends.append(e)
tails.append(e[-1:])
if p == '' or p == '-':
for aa in inv_alph:
if aa not in tails:
self._signed_ends[a].append(p_opp + aa)
else:
self._signed_ends[a].append(p)
for e in removed_ends:
self._signed_ends[a].remove(e)
if prefix_counts[
p] == N - 2 and p_opp in self._signed_ends[a]:
removed_end = True
for e in self._signed_ends[a]:
if e[:-1] == p:
removed_ends.append(e)
tails.append(e[-1:])
tails.append(inv_alph.inverse_letter(p[-1:]))
for aa in inv_alph:
if aa not in tails:
self._signed_ends[a].append(p_opp + aa)
for e in removed_ends:
self._signed_ends[a].remove(e)
self._signed_ends[a].remove(p_opp)
def _build_endmap(self,consolidate=False):
"""
Builds the map on ends for graph_map. Called when Core is initialized.
"""
alph=self._graph_map.domain()._alphabet
inv_alph=self._inv_graph_map.domain()._alphabet
FA=FreeGroup(alph)
FB=FreeGroup(inv_alph)
for x in self._graph_map.domain().edge_labels():
X=alph.inverse_letter(x)
self._signed_ends[x]=[]
self._signed_ends[X]=[]
for x in self._inv_graph_map.domain().edge_labels():
X=inv_alph.inverse_letter(x)
inv_image_x = self._inv_graph_map(x)
vp='' # vanishing path
for a in inv_image_x:
A=alph.inverse_letter(a)
if vp=='' or vp[-1]!=X: # coming from + side, no cancellation
self._signed_ends[a].append(vp + x)
else: # coming from - side, cancels x
self._signed_ends[a].append('-' + vp) # - + vp #
# add to vanishing path
vp=str(FB.reduced_product(str(self._graph_map(A)),vp))
# repeat for inverse, we've added to the vanishing path because we are
# coming from the other side
if vp=='' or vp[-1]!=X: # coming from - side, no cancellation
self._signed_ends[A].append('-' + vp + x) # - + vp + x
else: # coming from + side, cancels x
self._signed_ends[A].append(vp)
if consolidate:
for a in self._signed_ends.keys():
removed_end=True
N=2*inv_alph.cardinality() # should take into account the actual vertex...
while removed_end:
removed_end=False
prefix=[]
for e in self._signed_ends[a]:
prefix.append(e[:-1])
prefix_counts=dict((p,prefix.count(p)) for p in set(prefix))
for p in prefix_counts.keys():
p_opp = p[1:] if p[:1]=='-' else '-'+p
removed_ends=[]
tails=[]
if prefix_counts[p]==N-1:
removed_end=True
for e in self._signed_ends[a]:
if e[:-1]==p:
removed_ends.append(e)
tails.append(e[-1:])
if p=='' or p=='-':
for aa in inv_alph:
if aa not in tails: self._signed_ends[a].append(p_opp+aa)
else: self._signed_ends[a].append(p)
for e in removed_ends: self._signed_ends[a].remove(e)
if prefix_counts[p]==N-2 and p_opp in self._signed_ends[a]:
removed_end=True
for e in self._signed_ends[a]:
if e[:-1]==p:
removed_ends.append(e)
tails.append(e[-1:])
tails.append(inv_alph.inverse_letter(p[-1:]))
for aa in inv_alph:
if aa not in tails: self._signed_ends[a].append(p_opp+aa)
for e in removed_ends: self._signed_ends[a].remove(e)
self._signed_ends[a].remove(p_opp)
def tribonacci():
"""
Tribonacci automorphism.
"""
return FreeGroupAutomorphism("a->ab,b->ac,c->a", FreeGroup(3))
def inverse(self):
"""A homotopy inverse of ``self``.
For ``t1=self.domain().spanning_tree()`` and
``t2=self.codomain().spanning_tree()``. The alphabet ``A`` is
made of edges not in ``t1`` identified (by their order in the
alphabets of the domain and codomain) to letters not in
``t2``. The automorphism ``phi`` of ``FreeGroup(A)`` is
defined using ``t1`` and ``t2``. The inverse map is given by
``phi.inverse()`` using ``t1`` and edges from ``t2`` are
mapped to a single point (the root of ``t1``).
In particular the inverse maps all vertices to the root of ``t1``.
WARNING:
``self`` is assumed to be a homotopy equivalence.
"""
from free_group import FreeGroup
from free_group_automorphism import FreeGroupAutomorphism
G1 = self.domain()
A1 = G1.alphabet()
t1 = G1.spanning_tree()
G2 = self.codomain()
A2 = G2.alphabet()
t2 = G2.spanning_tree()
A = AlphabetWithInverses(len(A1) - len(G1.vertices()) + 1)
F = FreeGroup(A)
map = dict()
translate = dict()
i = 0
for a in A1.positive_letters():
l = len(t1[G1.initial_vertex(a)]) - len(t1[G1.terminal_vertex(a)])
if (l!=1 or t1[G1.initial_vertex(a)][-1]!=A1.inverse_letter(a)) and\
(l!=-1 or t1[G1.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
map[A[i]] = self(t1[G1.initial_vertex(a)] * Word([a]) *
G1.reverse_path(t1[G1.terminal_vertex(a)]))
translate[A[i]] = a
translate[A.inverse_letter(A[i])] = A1.inverse_letter(a)
i += 1
rename = dict()
edge_map = dict()
i = 0
for a in A2.positive_letters():
l = len(t2[G2.initial_vertex(a)]) - len(t2[G2.terminal_vertex(a)])
if (l!=1 or t2[G2.initial_vertex(a)][-1]!=A2.inverse_letter(a)) and\
(l!=-1 or t2[G2.terminal_vertex(a)][-1]!=a): # a is not in the spanning tree
rename[a] = A[i]
rename[A2.inverse_letter(a)] = A.inverse_letter(A[i])
i += 1
else:
edge_map[a] = Word()
for a in map:
map[a] = F([rename[b] for b in map[a] if b in rename])
phi = FreeGroupAutomorphism(map, F)
psi = phi.inverse()
i = 0
for a in A2.positive_letters():
if a not in edge_map:
result = Word()
for b in psi.image(A[i]):
c = translate[b]
result = result * t1[G1.initial_vertex(c)] * Word(
[c]) * G1.reverse_path(t1[G1.terminal_vertex(c)])
edge_map[a] = G1.reduce_path(result)
i += 1
return GraphMap(G2, G1, edge_map)