Out(Fn)

In mathematics, Out(F_n) is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.

Outer space

Out(F_n) acts geometrically on a cell complex known as Culler–Vogtmann Outer space, which can be thought of as the Teichmüller space for a bouquet of circles.

Definition

A point of the outer space is essentially an R-graph X homotopy equivalent to a bouquet of n circles together with a certain choice of a free homotopy class of a homotopy equivalence from X to the Bouquet of n circles. An R-graph is just a weighted Graph with weights in R. The sum of all weights should be 1 and all weights should be positive. To avoid ambiguity (and to get a finite dimensional space) it is furthermore required that the valency of each vertex should be at least 2.

A more descriptive view avoiding the homotopy equivalence f is the following. We may fix an identification of the fundamental group of the bouquet of n circles with the free group F_n in n variables. Furthermore we may choose a maximal tree in X and choose for each remaining edge a direction. We will now assign to each remaining edge e a word in F_n in the following way. Consider the closed path starting with e and then going back to the origin of e in the maximal tree. Composing this path with f we get a closed path in a bouquet of n circles and hence an element in its fundamental group F_n. This element is not well defined; if we change f by a free homotopy we obtain another element. It turns out, that those two elements are conjugate to each other, and hence we can choose the unique cyclically reduced element in this conjugacy class. It is possible to reconstruct the free homotopy type of f from these data. This view has the advantage, that it avoids the extra choice of f and has the disadvantage that additional ambiguity arises, because one has to choose a maximal tree and an orientation of the remaining edges.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Out(Fn)

Outer Space

by Swateezy
Outer Space

by Mob Rules
Outer Space

by System Of A Down
Outer Space

by The Muffs
Outer Space

by Yung Berg
Outer Space

by Freezepop
Outer Space

by Until June
Outer Space

by John Grant
Outer Space

by Venetian Princess
Outer Space

by Ace Frehley
Outer Space

by Dead Flowers
Outer Space

by Danny Brown
Outer Space

by Muffs

Outer Space

by: Swateezy

Feat. KJ and Lil Won
Bitch we outta space
Smokin on kush don't ask me why smell the loud when you passin by come sit on mars with this astrounaut
Bitch we outta space
Smokin floatin loaded speakin in slow motion I can barely even focus
Bitch we outta space
[KJ]
I'm on a kush cloud pow
Nigga smokin dat loud wow
Hoes flock to my squad because they see the drugs we around that's how we get down astronaut clique gotta flawless style we do it grande fam cigarillos get broke down light the kush baseball da blunt homerun smoker sammy sosa I got that kush I thought you knew cuz
One day we on clouds me won swat das how we get down posted loaded floatin an these chicks started smelling that loud approaching why woos like can we join
I'm like you gotta set of titties hit tht blunt won stupid faded swat stupid faded these chicks was dtf so you kno how we played it the slips one nation an I'm like mannnn
We da astrounaut clique
An we bout to take off nigga
Make yo chick my astrounaut bitch
Always on mary jane nigga
We da astronaut clique so respect my mind nigga the slip takin over shit an I'm like mannnnn
Bitch we outta space
Smokin on kush don't ask me why smell the loud when you passin by come sit on mars with this astronaut
Bitch we outta space
Smokin floatin loaded speakin in slow motion I can barely even focus
Bitch we outta space
[Won]
Whoo I'm high mannn I'm loaded
I'm floating steady smoking on potent potion not fokus on where a nigga going
So I jux sit in a daze headed ta space don't judge my ways
I'm On dat plane witcho main nd she giving me face
U walk by nd u smell da kush nd look & stare n amazement
Kuz it's loud as hell nd u kant tell either if I'm black or I'm Asian
Kuz.My eyes low like I'm on hydro
But I'm not I'm jux on kloud 9
Where ever I go u know I blow on kouple grams up to a ounce
I'm gon ima astronaut sooo don't ask me why
R u tryna choke on this purple loud if not u need to move the fuck around
Big chief on dis dodo slo tho kuz ion really solo ya bozo
Klown lookin ass homo pop wit ya fofo bang bang like my name robo
So split it fill it twist it lick it then light that hoe up
Puff puff pass chug that gas that feeling is real huh
If I'm lifted I ain't trippn bcuz my life is too geechie
Bitch I'm outta space wit my brotha k nd my patna swateezy
Yea won
Bitch we outta space
Smokin on kush don't ask me why smell the loud when you passin by come sit on mars with this astronaut
Bitch we outta space
Smokin floatin loaded speakin in slow motion I can barely even focus
Bitch we outta space
[Swat]
I'm speakin in slow motion
Smoking on that potent
Weed loud it's a commotion
I'm feelin myself wheres the lotion
We got the plan in motion
Troy and clay they coachin
Burning bread we toastin
Eyes red and they loccing
There's no one in the city even close to fuckin with me
Yeah I'm fuckin dope like bobby and whitney
Puff puff pass and we smokin like a chimney
Kj won and a couple others with me
You must heard I'm hot dog gettin relish so you better katchup
Il take ya bitch and pass the blunt
Like a stuffy nose she backin up
Man these hoes be actin up
But I got them in check casanova shit
She get hard knocked that hova shit
We in the stars supernova shit
Kj I swear this the shit that we been made for
Talk is cheap so imma show them what they paid for
I spread her like a butter knife while you tryna spoon
But I'm forkin, twistin like contortion
She hot, scorching tongue all on my forskin
Fuck her then I'm out this bitch, like an abortion

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