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Register a new userRewriting systems, plain groups, and geodetic graphs
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We prove that a group is presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 if and only if the group is plain. Our proof goes via a new result concerning properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory.
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We show that groups presented by inverse-closed finite convergent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to the free product of finitely many finite groups and copies of $mathbb Z$) if and only if a certain relation on the set of non-trivial finite-order elements of the group is transitive on a bounded set. We use this to prove that deciding if a group presented by an inverse-closed finite convergent length-reducing rewriting system is not plain is in $mathsf{NP}$. A yes answer would disprove a longstanding conjecture of Madlener and Otto from 1987. We also prove that the isomorphism problem for plain groups presented by inverse-closed finite convergent length-reducing rewriting systems is in $mathsf{PSPACE}$.
Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. By Olshanskii-Osin-Sapir, that property is weaker than the property of having Morse (rank 1) quasi-geodesics. Using our characterization of Morse quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur that states that mapping class groups cannot contain copies of irreducible lattices in semi-simple Lie groups of higher ranks. It also gives a generalization of the result of Birman-Lubotzky-McCarthy about solvable subgroups of mapping class groups not covered by the Tits alternative of Ivanov and McCarthy. We show that any group acting acylindrically on a simplicial tree or a locally compact hyperbolic graph always has many periodic Morse quasi-geodesics (i.e. Morse elements), so its divergence functions are never linear. We also show that the same result holds in many cases when the hyperbolic graph satisfies Bowditchs properties that are weaker than local compactness. This gives a new proof of Behrstocks result that every pseudo-Anosov element in a mapping class group is Morse. On the other hand, we conjecture that lattices in semi-simple Lie groups of higher rank always have linear divergence. We prove it in the case when the $mathbb{Q}$-rank is 1 and when the lattice is $SL_n(mathcal{O}_S)$ where $nge 3$, $S$ is a finite set of valuations of a number field $K$ including all infinite valuations, and $mathcal{O}_S$ is the corresponding ring of $S$-integers.
We define Poincar{e} profiles of Dirichlet type for graphs of bounded degree, in analogy with the Poincar{e} profiles (of Neumann type) defined in [HMT19]. The obvious first definition yields nothing of interest, but an alternative definition yields a spectrum of profiles which are quasi-isometry invariants and monotone with respect to subgroup inclusion. Moreover, in the extremal cases $p=1$ and $p=infty$, they detect the Fo lner function and the growth function respectively.
We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2epsilon)^{1/2p}$-uniformly Lipschitz) with $p$ varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal $p$ for which $L^p$-fixed point properties hold and the conformal dimension of the boundary. In the Gromov density model, we prove that for every $p_0 in [2, infty)$ for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on $L^p$-spaces that are $(2-2epsilon)^{1/2p}$-uniformly Lipschitz, and this for every $pin [2,p_0]$. To accomplish these goals we find new bounds on the first eigenvalue of the p-Laplacian on random graphs, using methods adapted from Kahn and Szemeredis approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to $L^p$-spaces previous results for Kazhdans Property (T) established by Zuk and Ballmann-Swiatkowski.
Let $G$ be a finite group and $sigma$ a partition of the set of all? primes $Bbb{P}$, that is, $sigma ={sigma_i mid iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_i$ and $sigma_icap sigma_j= emptyset $ for all $i e j$. If $n$ is an integer, we write $sigma(n)={sigma_i mid sigma_{i}cap pi (n) e emptyset }$ and $sigma (G)=sigma (|G|)$. We call a graph $Gamma$ with the set of all vertices $V(Gamma)=sigma (G)$ ($G e 1$) a $sigma$-arithmetic graph of $G$, and we associate with $G e 1$ the following three directed $sigma$-arithmetic graphs: (1) the $sigma$-Hawkes graph $Gamma_{Hsigma }(G)$ of $G$ is a $sigma$-arithmetic graph of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{Hsigma }(G))$ if $sigma_jin sigma (G/F_{{sigma_i}}(G))$; (2) the $sigma$-Hall graph $Gamma_{sigma Hal}(G)$ of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{sigma Hal}(G))$ if for some Hall $sigma_i$-subgroup $H$ of $G$ we have $sigma_jin sigma (N_{G}(H)/HC_{G}(H))$; (3) the $sigma$-Vasilev-Murashko graph $Gamma_{{mathfrak{N}_sigma }}(G)$ of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{{mathfrak{N}_sigma}}(G))$ if for some ${mathfrak{N}_{sigma }}$-critical subgroup $H$ of $G$ we have $sigma_i in sigma (H)$ and $sigma_jin sigma (H/F_{{sigma_i}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.
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