Ein neuer Beweis eines Satzes von Magnus

The author provides a new and very simple proof of the following theorem of Magnus (2): Theorem. Let the group G have both a presentation with n + r generators and r relations also a presentation with n generators. Then G is the free group on n generator

Groups with infinite homology

No abstract available

Mitochondrial phylogeography of baboons (Papio spp.) – Indication for introgressive hybridization?

<p>Abstract</p> <p>Background</p> <p>Baboons of the genus <it>Papio </it>are distributed over wide ranges of Africa and even colonized parts of the Arabian Peninsula. Traditionally, five phenotypically distinct species are recognized, but recent molecular studies were not able to resolve their phylogenetic relationships. Moreover, these studies revealed para- and polyphyletic (hereafter paraphyletic) mitochondrial clades for baboons from eastern Africa, and it was hypothesized that introgressive hybridization might have contributed substantially to their evolutionary history. To further elucidate the phylogenetic relationships among baboons, we extended earlier studies by analysing the complete mitochondrial cytochrome <it>b </it>gene and the 'Brown region' from 67 specimens collected at 53 sites, which represent all species and which cover most of the baboons' range.</p> <p>Results</p> <p>Based on phylogenetic tree reconstructions seven well supported major haplogroups were detected, which reflect geographic populations and discordance between mitochondrial phylogeny and baboon morphology. Our divergence age estimates indicate an initial separation into southern and northern baboon clades 2.09 (1.54–2.71) million years ago (mya). We found deep divergences between haplogroups within several species (~2 mya, northern and southern yellow baboons, western and eastern olive baboons and northern and southern chacma baboons), but also recent divergence ages among species (< 0.7 mya, yellow, olive and hamadryas baboons in eastern Africa).</p> <p>Conclusion</p> <p>Our study confirms earlier findings for eastern Africa, but shows that baboon species from other parts of the continent are also mitochondrially paraphyletic. The phylogenetic patterns suggest a complex evolutionary history with multiple phases of isolation and reconnection of populations. Most likely all these biogeographic events were triggered by multiple cycles of expansion and retreat of savannah biomes during Pleistocene glacial and inter-glacial periods. During contact phases of populations reticulate events (i.e. introgressive hybridization) were highly likely, similar to ongoing hybridization, which is observed between East African baboon populations. Defining the extent of the introgressive hybridization will require further molecular studies that incorporate additional sampling sites and nuclear loci.</p