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.
Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $G$. A well-known conjecture of D. Gluck claims that if $G$ is solvable then $|G:F(G)|leq b(G)^{2}$. We confir
m this conjecture in the case where $|F(G)|$ is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.
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 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
cible representations and complex of groups decompositions. In this paper, we give a specific condition on Coxeter presentations that implies FA_n and show that this condition is in fact equivalent to FA_n for n=1 and 2. As part of the proof, we compute the Gersten-Stallings angles between special subgroups of Coxeter groups.
We confirm the Hanna Neumann conjecture for topologically finitely generated closed subgroups $U$ and $W$ of a nonsolvable Demushkin group $G$. Namely, we show that begin{equation*} sum_{g in U backslash G/W} bar d(U cap gWg^{-1}) leq bar d(U) bar d(
W) end{equation*} where $bar d(K) = max{d(K) - 1, 0}$ and $d(K)$ is the least cardinality of a topological generating set for the group $K$.