Virtually splitting the map from Aut(G) to Out(G)

We give an elementary criterion on a group G for the map from Aut(G) to Out(G) to split virtually. This criterion applies to many residually finite CAT(0) groups and hyperbolic groups, and in particular to all finitely generated Coxeter groups. As a consequence the outer automorphism group of any finitely generated Coxeter group is residually finite and virtually torsion-free.Comment: 10 pages, 1 figur

The automorphism group of accessible groups

In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by taking extensions of relative automorphism groups of vertex groups, groups of Dehn twists and groups of automorphisms of free products. We apply this description and obtain a criterion for Out(G) to be finitely presented, as well as a necessary and sufficient condition for Out(G) to be finite. Consequences for hyperbolic groups are discussed.Comment: 18 pages, 3 figures. Section 4 rewritten and corrected, added reference

Locally compact convergence groups and n-transitive actions

All sigma-compact, locally compact groups acting sharply n-transitively and continuously on compact spaces M have been classified, except for n=2,3 when M is infinite and disconnected. We show that no such actions exist for n=2 and that these actions for n=3 coincide with the action of a hyperbolic group on a space equivariantly homeomorphic to its hyperbolic boundary. We further give a characterization of non-compact groups acting 3-properly and transitively on infinite compact sets as non-elementary boundary transitive hyperbolic groups. The main tool is a generalization to locally compact groups of Bowditch's topological characterization of hyperbolic groups. Finally, in contrast to the case n=3, we show that for n>3, if a locally compact group acts continuously, n-properly and n-cocompactly on a locally connected metrizable compactum M, then M has a local cut point

Spectral rigidity of automorphic orbits in free groups

It is well-known that a point $T\in cv_N$ in the (unprojectivized) Culler-Vogtmann Outer space $cv_N$ is uniquely determined by its \emph{translation length function} $||.||_T:F_N\to\mathbb R$. A subset $S$ of a free group $F_N$ is called \emph{spectrally rigid} if, whenever $T,T'\in cv_N$ are such that $||g||_T=||g||_{T'}$ for every $g\in S$ then $T=T'$ in $cv_N$. By contrast to the similar questions for the Teichm\"uller space, it is known that for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$. In this paper we prove that for $N\ge 3$ if $H\le Aut(F_N)$ is a subgroup that projects to an infinite normal subgroup in $Out(F_N)$ then the $H$-orbit of an arbitrary nontrivial element $g\in F_N$ is spectrally rigid. We also establish a similar statement for $F_2=F(a,b)$, provided that $g\in F_2$ is not conjugate to a power of $[a,b]$. We also include an appended corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of $Out(F_N)$. Our original proof of Lemma 5.1 relied on a subgroup classification result of Handel-Mosher, originally stated by Handel-Mosher for arbitrary subgroups $H\le Out(F_N)$. After our paper was published, it turned out that the proof of the Handel-Mosher subgroup classification theorem needs the assumption that $H$ be finitely generated. The corrigendum provides an alternative proof of Lemma~5.1 which uses the corrected, finitely generated, version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of $Out(F_N)$ on the free factor complex (due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the corrigendum.Comment: Included a corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of Out(F_N). Note that, because of the arXiv rules, the corrigendum and the original article are amalgamated into a single pdf file, with the corrigendum appearing first, followed by the main body of the original articl

The arthritis-associated HLA-B*27:05 allele forms more cell surface B27 dimer and free heavy chain ligands for KIR3DL2 than HLA-B*27:09

Objectives. HLA-B*27:05 is associated with AS whereas HLA-B*27:09 is not associated. We hypothesized that different interactions with KIR immune receptors could contribute to the difference in disease association between HLA-B*27:05 and HLAB*27:09. Thus, the objective of this study was to compare the formation of β2m-free heavy chain (FHC) including B27 dimers (B272) by HLA-B*27:05 and HLA-B*27:09 and their binding to KIR immunoreceptors. Methods. We studied the formation of HLA-B*27:05 and HLA-B*27:09 heterotrimers and FHC forms including dimers in vitro and in transfected cells. We investigated HLA-B*27:05 and HLA-B*27:09 binding to KIR3DL1, KIR3DL2 and LILRB2 by FACS staining with class I tetramers and by quantifying interactions with KIR3DL2CD3ε-reporter cells and KIR3DL2-expressing NK cells. We also measured KIR expression on peripheral blood NK and CD4 T cells from 18 HLA-B*27:05 AS patients, 8 HLA-B27 negative and 12 HLA-B*27:05+ and HLA-B*27:09+ healthy controls by FACS staining. Results. HLA-B*27:09 formed less B272 and FHC than HLA-B*27:05. HLA-B*27:05-expressing cells stimulated KIR3DL2CD3ε-reporter T cells more effectively. Cells expressing HLA-B*27:05 promoted KIR3DL2+ NK cell survival more strongly than HLA-B*27:09. HLA-B*27:05 and HLA-B*27:09 dimer tetramers stained KIR3DL1, KIR3DL2 and LILRB2 equivalently. Increased proportions of NK and CD4 T cells expressed KIR3DL2 in HLA-B*27:05+ AS patients compared with HLA-B*27:05+, HLA-B*27:09+ and HLA-B27− healthy controls. Conclusion. Differences in the formation of FHC ligands for KIR3DL2 by HLA-B*27:05 and HLA-B*27:09 could contribute to the differential association of these alleles with A