Mixed quasi-\'etale surfaces, new surfaces of general type with pg=0p_g=0 and their fundamental group

We call a projective surface $X$ mixed quasi-\'etale quotient if there exists a curve $C$ of genus $g(C)\geq 2$ and a finite group $G$ that acts on $C\times C$ exchanging the factors such that $X=(C\times C)/G$ and the map $C\times C \rightarrow X$ has finite branch locus. The minimal resolution of its singularities is called mixed quasi-\'etale surface. We study the mixed quasi-\'etale surfaces under the assumption that $(C\times C)/G^0$ has only nodes as singularities, where $G^0\triangleleft G$ is the index two subgroup of the elements that do not exchange the factors. We classify the minimal regular surfaces with $p_g=0$ whose canonical model is a mixed quasi-\'etale quotient as above. All these surfaces are of general type and as an important byproduct, we provide an example of a numerical Campedelli surface with topological fundamental group \bbZ_4, and we realize 2 new topological types of surfaces of general type. Three of the families we construct are \bbQ-homology projective planes.Comment: 18 pages, 3 tables, v2: change title, exposition improved; v3: minor corrections, final version to be published in Collectanea Mathematic

On the reduction of Alperin's Conjecture to the quasi-simple groups

We show that the refinement of Alperin's Conjecture proposed in "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, can be proved by checking that this refinement holds on any central k*-extension of a finite group H containing a normal simple group S with trivial centralizer in H and p'-cyclic quotient H/S. This paper improves our result in [ibidem, Theorem 16.45] and repairs some bad arguments there

Decomposing locally compact groups into simple pieces

We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple. Two appendices introduce results and examples around the concept of quasi-product.Comment: Index added; minor change