相关论文: Averaged Dehn Functions for Nilpotent Groups

This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme…

We prove foundational results for homological Dehn functions of groups of type $FP_2$ such as superadditivity and the invariance under quasi-isometry. We then study the homological Dehn functions of Leary's groups $G_L(S)$ providing methods…

We construct easy embeddings of relatively free groups (say the free Burnside group, the free solvable group) into finitely presented groups. We introduce a concept of verbal isoperimetric function of a group variety. We prove that if the…

The functions $F_{G}(n)$ measures the asymptotic behavior of residual finiteness for a finitely generated group $G$. In previous work \cite{Pengitore_1}, the author claimed a characterization for $F_{N}(n)$ when $N$ is a finitely generated…

Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to…

Suppose $\Gamma$ is an arithmetic group defined over a global field $K$, that the $K$-type of $\Gamma$ is $A_n$ with $n \geq 2$, and that the ambient semisimple group that contains $\Gamma$ as a lattice has at least two noncocompact…

Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\log{\mathcal{L}(s)}$ and $\mathcal{L}'(s)/\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann, Dirichlet and…

Given a finitely presented group $G$ and a surjective homomorphism $G\to \mathbb{Z}^n$ with finitely presented kernel $K$, we give an upper bound on the Dehn function of $K$ in terms of an area-radius pair for $G$. As a consequence we…

We construct examples of finitely presented simple groups whose Dehn functions are at least exponential. To the best of our knowledge, these are the first such examples known. Our examples arise from R\"over-Nekrashevych groups, using…

We prove that every finitely generated nilpotent group of class c admits a polynomial isoperimetric function of degree c+1 and a linear upper bound on its filling length function.

Addressing a question of Gromov, we give a rate in Pansu's theorem about the convergence in Gromov-Hausdorff metric of a finitely generated nilpotent group equipped with a left-invariant word metric scaled by a factor 1/n towards its…

The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for "generic" periodic second-order elliptic operators on noncompact Riemannian co-compact coverings with abelian…

V.S.Guba had proved that the R.Thompson group $T$ satisfies polynomial isoperimetric inequality and $\Phi_T(n)\preceq n^7$, where $\Phi_T$ is the Dehn function of group $T$. In this paper, we show that $\Phi_T(n)\preceq n^5$.

Let $G$ be a random torsion-free nilpotent group generated by two random words of length $\ell$ in $U_n(\mathbb{Z})$. Letting $\ell$ grow as a function of $n$, we analyze the step of $G$, which is bounded by the step of $U_n(\mathbb{Z})$.…

In this article we deduce an analogue of Quillen's Local-Global Principle for the elementary subgroup of the general quadratic group and the hermitian group. We show that the unstable K_1-groups of the hermitian groups are nilpotent by…

Given a finitely generated, torsion-free nilpotent group, we find the maximum possible (critical) regularity for its faithful actions by diffeomorphisms of the closed or half-open interval and of the circle. Our result gives an expression…

We point out an asymptotic formula for the power moments of the function $a(n)$, representing the number of non-isomorphic Abelian groups of order $n$. For the quadratic moment this improves an earlier result due to L. Zhang, M. L\"u and W.…

In this article we relate word and subgroup growth to certain functions that arise in the quantification of residual finiteness. One consequence of this endeavor is a pair of results that equate the nilpotency of a finitely generated group…

Gromov asked whether an asymptotic cone of a finitely generated group was always simply connected or had uncountable fundamental group. We prove that Gromov's dichotomy holds for asympotic cones with cut points, as well as, HNN extensions…

We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise…