For a finite alphabet A and shift X ⊆ AZ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group Aut(X). For such systems, we show that every finitely generated subgroup of Aut(X) is virtually Zd, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if X is tr...

Let X be an irreducible complete nonsingular curve of genus g over an algebraically closed field k of positive characteristic p. If g > 2, the automorphism group Aut(X) of X is known to be a finite group, and moreover its order is bounded from above by a polynomial in g of degree four (Stichtenoth). In this paper we consider the p-rank -7 of X and investigate relations between if and Aut(X). We...

0.1. Results. We study the automorphisms of some generalizations of Thompson’s groups and their underlying structures. The automorphism groups of two of Thompson’s original groups were analyzed in [2] and were shown to be “small” and “unexotic.” Our results differ sharply from [2] in that we show that the automorphism groups of the generalizations are “large” and have “exotic” elements. The ter...

Autism (AUT), schizophrenia (SCZ) and bipolar disorder (BPD) are three highly heritable neuropsychiatric conditions. Clinical similarities and genetic overlap between the three disorders have been reported; however, the causes and the downstream effects of this overlap remain elusive. By analyzing transcriptomic RNA-sequencing data generated from post-mortem cortical brain tissues from AUT, SCZ...

Let X be a right-angled building. We show that the lattices in Aut(X) share many properties with tree lattices. For example, we characterise the set of covolumes of uniform and of nonuniform lattices in Aut(X), and show that the group Aut(X) admits an infinite ascending tower of uniform and of nonuniform lattices. These results are proved by constructing a functor from graphs of groups to compl...

Let X $X$ be a compact complex space in Fujiki's Class C $\mathcal {C}$ . We show that the group Aut ( ) $\operatorname{Aut}(X)$ of all biholomorphic automorphisms has Jordan property: there is (Jordan) constant J = $J J(X)$ such any finite subgroup G ⩽ $G\leqslant \operatorname{Aut}(X)$ an abelian H $H\leqslant G$ with index [ : ] $[G:H]\leqslant J$ This extends, quite different method, result...

When Aut(A) and Homeo(Prim(A)) are homeomorphic, where A is a C *-algebra Abstract In this paper, we discuss when Aut(A) and Homeo(Prim(A)) are homeomorphic, where A is a C *-algebra. 1 Preliminaries Let A be C *-algebra. Then the collection Aut(A) of ⋆-automorphisms of A is a group under composition. We give Aut(A) the point-norm topology, that is, α n → α if and only if α n (a) → α(a) for all...

The stabilization of Hi(Aut(G);Z) for i large has been shown to hold for most groups by the main theorem of [3] and [9, Corollary 1.3].1 The stabilization of Hi(Aut(G);Z) in contrast has so far only been known in two extreme cases: when G is abelian and when G has trivial center and does not factorize as a direct product. Indeed, in the first case Aut(Gn) is isomorphic to GLn(End(G)), which is ...

We prove a characterization theorem for the unit polydisc ∆ ⊂ C in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut(M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut(M) and Aut(∆) are isomorphic as topological groups equipped with the compact-open topolo...

We study the complex of partial bases of a free group, which is an analogue for Aut(Fn) of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of Aut(Fn) is highly connected. Using these results, we give a new, topological proof of a theorem of Magnus that asserts that the Torelli subgroup...