Let $G$ be a semisimple Lie group that is not isogenous to the isometry group of real or complex hyperbolic $n$-space. We show that the rational rank is a profinite invariant for irreducible lattices in $G$, unless $G$ has type $D_{2n}$ which forms a genuine exception. In particular, profinitely isomorphic lattices in $G$ of type $D_{2n}$ can have differing virtual cohomological dimension. As another corollary, uniformity is a profinite property for irreducible lattices in $G$ outside type $D_4$. Assuming CSP, the existence of triality symmetries gives uniform and non-uniform profinitely isomorphic lattices in type $D_4$.
Let $X$ be an algebraic variety of dimension $d$ over an algebraically closed field $k$ of zero characteristic. Suppose that $G$ is a virtually polycyclic subgroup in the group $\mathrm{Bir}(X)$ of birational automorphisms of $X$. We show that the virtual derived length of $G$ does not exceed $2d+1$. Moreover, if $X$ i...
We show that every faithful, transitive, and generically $(n+2)$-transitive action of a connected group $G$ on an irreducible variety $X$ of dimension $n>0$, all defined over an algebraically closed field $F$, is isomorphic to the natural action of the projective linear group $PGL_{n+1}(F)$ on the projective space $\ma...
Let $k$ be an algebraically closed field of characteristic $p>0$, and let $S$ be a minimal smooth projective surface of general type over $k$. If the second Chern class satisfies $c_2(S)<0$, then the order of the automorphism group satisfies $|\mathrm{Aut}_k(S)|<2598(K_S^2)^4$, and the order of every abelian subgroup $...
Let $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. This answers a conjecture by Erdmann, Kl\'asz...
For a finite group $G$, let $\lambda(G)$ denote the minimum number of orbits on the elements of a finite lattice $L$ with $\operatorname{Aut}(L)\cong G$. Babai and Goodman conjectured that $\lambda(G)$ is bounded by an absolute constant. We prove that $\lambda(G)\leq 50$ for every finite group $G$, thereby confirming t...
Jia-Li Du, Andrea Lucchini, Joy Morris et al.· 0 citations
For a group $G$, let $a(G)$ denote the minimum number of abelian subgroups covering $G$, and let $\omega(G)$ be the maximum size of a pairwise noncommuting set. We prove the exact formula $\sup_{\omega(G)\leq n}a(G)=2^{\lfloor(n-1)/2\rfloor}+1$ for every $n\geq 63$. Equality holds precisely for groups with derived subg...
Guillaume Lecomte· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
