For any right-angled Artin group AΓ we construct a finite-dimensional space OΓ on which the group Out(AΓ) of outer automorphisms of AΓ acts properly. We prove that OΓ is contractible, so that the quotient is a rational classifying space for Out(AΓ). The space OΓ blends features of the symmetric space of lattices in R n with those of Outer space for the free group Fn. Points in OΓ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with AΓ.
We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A Γ ), for certain right-angled Artin groups A Γ . This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A Γ )). This demonstrates a marked departure from the known behavior of Out(Out(A Γ )) when A Γ is free or free abelian, as in these cases Out(Out(A Γ )) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite order members of certain Out(Aut(A Γ )).
OBJECTIVEBenign external hydrocephalus (BEH) is an enlargement of the subarachnoid spaces (SASs) that can be seen in young children. It is controversial whether children with BEH are predisposed to developing subdural hemorrhage (SDH) with or without trauma. This issue is clinically relevant as a finding of unexplained SDH raises concerns about child abuse and often prompts child protection and law enforcement investigations.METHODSThis retrospective study included children (1–24 months of age) who underwent head CT scanning after an accidental fall of less than 6 feet. Head CT scans were reviewed, cranial findings were documented, and the SAS was measured and qualitatively evaluated. Enlarged SAS was defined as an extraaxial space (EAS) greater than 4 mm on CT scans. Clinical measurements of head circumference (HC) were noted, and the head circumference percentile was calculated. The relationship between enlarged SAS and HC percentile, and enlarged SAS and intracranial hemorrhage (ICH), were investigated using bivariate analysis.RESULTSOf the 110 children included in this sample, 23 had EASs greater than 4 mm. The mean patient age was 6.8 months (median 6.0 months). Thirty-four patients (30.9%) had ICHs, including subarachnoid/subpial (6.2%), subdural (6.2%), epidural (5.0%), and unspecified extraaxial hemorrhage (16.5%). Enlarged SAS was positively associated with subarachnoid/subpial hemorrhage; there was no association between enlarged SASs and either SDH or epidural hemorrhage. A larger SAS was positively associated with larger HC percentile; however, HC percentile was not independently associated with ICH.CONCLUSIONSEnlarged SAS was not associated with SDH, but was associated with other ICHs. The authors' findings do not support the theory that BEH predisposes children to SDH with minor accidental trauma.
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