ﻻ يوجد ملخص باللغة العربية
We prove Wises $W$-cycles conjecture. Consider a compact graph $Gamma$ immering into another graph $Gamma$. For any immersed cycle $Lambda:S^1to Gamma$, we consider the map $Lambda$ from the circular components $mathbb{S}$ of the pullback to $Gamma$. Unless $Lambda$ is reducible, the degree of the covering map $mathbb{S}to S^1$ is bounded above by minus the Euler characteristic of $Gamma$. As a consequence, we obtain a homological version of coherence for one-relator groups.
We prove a conjecture made by Gilman in 1984 that the groups presented by finite, monadic, confluent rewriting systems are precisely the free products of free and finite groups.
As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${rm PSL}_n(q)$ is prime. We present heuristic
Starting with a lattice with an action of $mathbb{Z}$ or $mathbb{R}$, we build a Helly graph or an injective metric space. We deduce that the $ell^infty$ orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard
We study global fixed points for actions of Coxeter groups on nonpositively curved singular spaces. In particular, we consider property FA_n, an analogue of Serres property FA for actions on CAT(0) complexes. Property FA_n has implications for irredu
We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even