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Register a new userAutomorphisms of non-spherical buildings have unbounded displacement
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Authors
Peter Abramenko
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If f is a nontrivial automorphism of a thick building Delta of purely infinite type, we prove that there is no bound on the distance that f moves a chamber. This has the following group-theoretic consequence: If G is a group of automorphisms of Delta with bounded quotient, then the center of G is trivial.
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Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion of cocharacter-closed $G(k)$-orbits in $V$. In earlier work we used a rationality condition on the point stabilizer of a $G$-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding $G(k)$-orbit in $V$. In the present paper we employ building-theoretic techniques to derive analogous results.
Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.
A rigid automorphism of a linking system is an automorphism which restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian, and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group $G$ restricts to the identity on the centric linking system for $G$, then it is of $p$-order modulo the group of inner automorphisms, provided $G$ has no nontrivial normal $p$-subgroups. We present two applications of this last result, one to tame fusion systems.
An odd Coxeter group $W$ is one which admits a Coxeter system $(W,S)$ for which all the exponents $m_{ij}$ are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs $mathcal{V}_{(W,S)}$ are trees. It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same multiset of finite exponents. In particular, each group in this family is isomorphic to a group that admits a Coxeter system whose associated labeled graph is a star shaped tree. We give the complete description of the automorphism group of this group, and derive a sufficient condition for the splitting of the automorphism group as a semi-direct product of the inner and the outer automorphism groups. As applications, we prove that Coxeter groups in this family satisfy the $R_infty$-property and are (co)-Hopfian. We compare structural properties, automorphism groups, $R_infty$-property and (co)-Hopfianity of a special odd Coxeter group whose only finite exponent is three with the braid group and the twin group.
Let $G$ be a finite group admitting a coprime automorphism $alpha$ of order $e$. Denote by $I_G(alpha)$ the set of commutators $g^{-1}g^alpha$, where $gin G$, and by $[G,alpha]$ the subgroup generated by $I_G(alpha)$. We study the impact of $I_G(alpha)$ on the structure of $[G,alpha]$. Suppose that each subgroup generated by a subset of $I_G(alpha)$ can be generated by at most $r$ elements. We show that the rank of $[G,alpha]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(alpha)$ has odd order, then $[G,alpha]$ has odd order too. Further, if every pair of elements from $I_G(alpha)$ generates a soluble, or nilpotent, subgroup, then $[G,alpha]$ is soluble, or respectively nilpotent.
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