291 research outputs found
On the number of outer automorphisms of the automorphism group of a right-angled Artin group
We show that there is no uniform upper bound on |Out(Aut(A))| when A ranges
over all right-angled Artin groups. This is in contrast with the cases where A
is free or free abelian: for all n, Dyer-Formanek and Bridson-Vogtmann showed
that Out(Aut(F_n)) = 1, while Hua-Reiner showed |Out(Aut(Z^n)| = |Out(GL(n,Z))|
< 5. We also prove the analogous theorem for Out(Out(A)). We establish our
results by giving explicit examples; one useful tool is a new class of graphs
called austere graphs
Threshold Graphs Maximize Homomorphism Densities
Given a fixed graph and a constant , we can ask what graphs
with edge density asymptotically maximize the homomorphism density of
in . For all for which this problem has been solved, the maximum is
always asymptotically attained on one of two kinds of graphs: the quasi-star or
the quasi-clique. We show that for any the maximizing is asymptotically
a threshold graph, while the quasi-clique and the quasi-star are the simplest
threshold graphs having only two parts. This result gives us a unified
framework to derive a number of results on graph homomorphism maximization,
some of which were also found quite recently and independently using several
different approaches. We show that there exist graphs and densities
such that the optimizing graph is neither the quasi-star nor the
quasi-clique, reproving a result of Day and Sarkar. We rederive a result of
Janson et al. on maximizing homomorphism numbers, which was originally found
using entropy methods. We also show that for large enough all graphs
maximize on the quasi-clique, which was also recently proven by Gerbner et al.,
and in analogy with Kopparty and Rossman we define the homomorphism density
domination exponent of two graphs, and find it for any and an edge
Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group
We define a family of groups that include the mapping class group of a genus
g surface with one boundary component and the integral symplectic group
Sp(2g,Z). We then prove that these groups are finitely generated. These groups,
which we call mapping class groups over graphs, are indexed over labeled
simplicial graphs with 2g vertices. The mapping class group over the graph
Gamma is defined to be a subgroup of the automorphism group of the right-angled
Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut
A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a theorem of
Magnus.Comment: 45 page
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