Automorphism orbit membership for right-angled Artin groups Matthew B. Day University of Arkansas matthewd@uark.edu GAGTA7@CCNY, May 30, 2013 Ar χ iv reference: 1211:0078, Full-featured peak reduction in right-angled Artin groups Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 1 / 22
Right-angled Artin groups Let Γ be a finite simplicial graph. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 2 / 22
Right-angled Artin groups Let Γ be a finite simplicial graph. We define a finitely presented group based on Γ: Definition The right-angled Artin group (RAAG) A Γ is the group with presentation: A Γ = � Vertices(Γ) |{ xy = yx | Γ has an edge from x to y }� . These are also called “graph groups” or “partially commutative groups” or “PC groups”. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 2 / 22
Right-angled Artin groups Let Γ be a finite simplicial graph. We define a finitely presented group based on Γ: Definition The right-angled Artin group (RAAG) A Γ is the group with presentation: A Γ = � Vertices(Γ) |{ xy = yx | Γ has an edge from x to y }� . These are also called “graph groups” or “partially commutative groups” or “PC groups”. So Γ encodes the commuting between generators of A Γ . Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 2 / 22
Right-angled Artin groups Let Γ be a finite simplicial graph. We define a finitely presented group based on Γ: Definition The right-angled Artin group (RAAG) A Γ is the group with presentation: A Γ = � Vertices(Γ) |{ xy = yx | Γ has an edge from x to y }� . These are also called “graph groups” or “partially commutative groups” or “PC groups”. So Γ encodes the commuting between generators of A Γ . These groups are simple to define but have interesting properties, especially concerning their subgroup structure, automorphism groups, and connections to geometry and topology. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 2 / 22
Examples of RAAGs Example: Γ edgeless implies A Γ ∼ = F n and Aut( A Γ ) ∼ = Aut( F n ). Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 3 / 22
Examples of RAAGs Example: Γ edgeless implies A Γ ∼ = F n and Aut( A Γ ) ∼ = Aut( F n ). = Z n and Aut( A Γ ) ∼ Example: Γ complete implies A Γ ∼ = GL( n , Z ). Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 3 / 22
Examples of RAAGs Example: Γ edgeless implies A Γ ∼ = F n and Aut( A Γ ) ∼ = Aut( F n ). = Z n and Aut( A Γ ) ∼ Example: Γ complete implies A Γ ∼ = GL( n , Z ). Free products of finitely many RAAGs are RAAGs (take disjoint unions of graphs). Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 3 / 22
Examples of RAAGs Example: Γ edgeless implies A Γ ∼ = F n and Aut( A Γ ) ∼ = Aut( F n ). = Z n and Aut( A Γ ) ∼ Example: Γ complete implies A Γ ∼ = GL( n , Z ). Free products of finitely many RAAGs are RAAGs (take disjoint unions of graphs). Direct products of finitely many RAAGs are RAAGs (take graph joins). Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 3 / 22
Examples of RAAGs Example: Γ edgeless implies A Γ ∼ = F n and Aut( A Γ ) ∼ = Aut( F n ). = Z n and Aut( A Γ ) ∼ Example: Γ complete implies A Γ ∼ = GL( n , Z ). Free products of finitely many RAAGs are RAAGs (take disjoint unions of graphs). Direct products of finitely many RAAGs are RAAGs (take graph joins). The group � a , b , c , d | [ a , b ] = [ b , c ] = [ c , d ] = 1 � is the RAAG of the following graph. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 3 / 22
Automorphisms of RAAGs Suppose Γ contains vertices x , y , z with x adjacent to y but y not adjacent to z . Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 4 / 22
Automorphisms of RAAGs Suppose Γ contains vertices x , y , z with x adjacent to y but y not adjacent to z . Then there is no automorphism φ ∈ Aut( A Γ ) with φ ( x ) = zx and φ ( v ) = v if v � = z for vertices v of Γ. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 4 / 22
Automorphisms of RAAGs Suppose Γ contains vertices x , y , z with x adjacent to y but y not adjacent to z . Then there is no automorphism φ ∈ Aut( A Γ ) with φ ( x ) = zx and φ ( v ) = v if v � = z for vertices v of Γ. Why? Because [ x , y ] = 1 but [ φ ( x ) , φ ( y )] � = 1. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 4 / 22
Automorphisms of RAAGs Suppose Γ contains vertices x , y , z with x adjacent to y but y not adjacent to z . Then there is no automorphism φ ∈ Aut( A Γ ) with φ ( x ) = zx and φ ( v ) = v if v � = z for vertices v of Γ. Why? Because [ x , y ] = 1 but [ φ ( x ) , φ ( y )] � = 1. Compare to Aut( F n ) and GL( n , Z ) where such automorphisms always exist and nearly generate the group (they are called Nielsen moves or transvections or elementary matrices ). Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 4 / 22
Automorphisms of RAAGs Suppose Γ contains vertices x , y , z with x adjacent to y but y not adjacent to z . Then there is no automorphism φ ∈ Aut( A Γ ) with φ ( x ) = zx and φ ( v ) = v if v � = z for vertices v of Γ. Why? Because [ x , y ] = 1 but [ φ ( x ) , φ ( y )] � = 1. Compare to Aut( F n ) and GL( n , Z ) where such automorphisms always exist and nearly generate the group (they are called Nielsen moves or transvections or elementary matrices ). Laurence’s theorem (1995) gives a concrete finite generating set for Aut( A Γ ) that can be read off of Γ. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 4 / 22
Automorphism orbits Let G be a group and let C G (or just C ) denote the set of conjugacy classes of G . Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 5 / 22
Automorphism orbits Let G be a group and let C G (or just C ) denote the set of conjugacy classes of G . Then Aut( G ) acts diagonally on both G k = G × · · · × G and C k G = C G × · · · × C G (for any k ). Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 5 / 22
Automorphism orbits Let G be a group and let C G (or just C ) denote the set of conjugacy classes of G . Then Aut( G ) acts diagonally on both G k = G × · · · × G and C k G = C G × · · · × C G (for any k ). Question (Whitehead problem): For a given group G , is there an algorithm that tells us whether two elements U and V in G k (or in C k G ) are in the same orbit? Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 5 / 22
Automorphism orbits Let G be a group and let C G (or just C ) denote the set of conjugacy classes of G . Then Aut( G ) acts diagonally on both G k = G × · · · × G and C k G = C G × · · · × C G (for any k ). Question (Whitehead problem): For a given group G , is there an algorithm that tells us whether two elements U and V in G k (or in C k G ) are in the same orbit? (That is, how can we tell if there is φ ∈ Aut( G ) with φ · U = V ?) Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 5 / 22
Automorphism orbits Let G be a group and let C G (or just C ) denote the set of conjugacy classes of G . Then Aut( G ) acts diagonally on both G k = G × · · · × G and C k G = C G × · · · × C G (for any k ). Question (Whitehead problem): For a given group G , is there an algorithm that tells us whether two elements U and V in G k (or in C k G ) are in the same orbit? (That is, how can we tell if there is φ ∈ Aut( G ) with φ · U = V ?) If we determine that U and V are in the same orbit, then we also want to produce φ ∈ Aut( G ) with φ · U = V . Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 5 / 22
Orbits of Matrix Groups,1 As a warm up, consider G = Q n and consider the action of Aut( G ) = GL( n , Q ) on k -tuples of elements of G : G k = ( Q n ) k = M n , k ( Q ), the set of n -row, k -column matrices. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 6 / 22
Orbits of Matrix Groups,1 As a warm up, consider G = Q n and consider the action of Aut( G ) = GL( n , Q ) on k -tuples of elements of G : G k = ( Q n ) k = M n , k ( Q ), the set of n -row, k -column matrices. The action is given by multiplying matrices from GL( n , Q ) on the left. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 6 / 22
Orbits of Matrix Groups,1 As a warm up, consider G = Q n and consider the action of Aut( G ) = GL( n , Q ) on k -tuples of elements of G : G k = ( Q n ) k = M n , k ( Q ), the set of n -row, k -column matrices. The action is given by multiplying matrices from GL( n , Q ) on the left. Then U , V ∈ M n , k ( Q ) are in the same orbit if and only if they have the same reduced row-echelon form. If so, we can find A ∈ GL( n , Q ) with A · U = V by row-reducing each. Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 6 / 22
Orbits of Matrix Groups,2 For G = Z n , the situation is trickier. Then Aut( G ) = GL( n , Z ) and G k = M n , k ( Z ). Matthew Day (U of Arkansas) Aut orbit membership for RAAGs May 30,2013 7 / 22
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