Related papers: Quantifying residual finiteness of arithmetic grou…
Normal residual finiteness growth measures how well a finitely generated group is approximated by its finite quotients. We show that any linear group $\Gamma \leq \mathrm{GL}_d(K)$ has normal residual finiteness growth asymptotically…
The solvable Farb growth of a group quantifies how well-approximated the group is by its finite solvable quotients. In this note we present a new characterization of polycyclic groups which are virtually nilpotent. That is, we show that a…
The residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. In this paper, we construct groups with arbitrarily large residual finiteness growth. We also demonstrate a new relationship…
If $g\in G$ is a non-trivial element in a residually finite group, then there exists by definition a finite group $Q$ and a homomorphism $\varphi: G \to Q$ such that $\varphi(g) \neq e$. The residual finiteness growth $\text{RF}_G$ of a…
A group $G$ is called residually finite if for every non-trivial element $g \in G$, there exists a finite quotient $Q$ of $G$ such that the element $g$ is non-trivial in the quotient as well. Instead of just investigating whether a group…
Residual finiteness growth gives an invariant that indicates how well-approximated a finitely generated group is by its finite quotients. We briefly survey the state of the subject. We then improve on the best known upper and lower bounds…
Let G be a Chevalley group scheme and B<=G a Borel subgroup scheme, both defined over Z. Let K be a global function field, S be a finite non-empty set of places over K, and O_S be the corresponding S-arithmetic ring. Then, the S-arithmetic…
For a residually finite group $G$, its normal subgroups $G\supset G_1\supset G_2\cdots$ with $\cap_{n\in\mathbb N}G_n=\{e\}$ and for a growth function $\gamma$ we construct a unitary representation $\pi_\gamma$ of $G$. For the minimal…
The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $\Gamma$ be a group and $\Delta$…
Residual finiteness growth measures how well-approximated a group is by its finite quotients. We prove that some related growth functions characterize linearity for a class of groups including all hyperbolic groups.
In this article we improve the known uniform bound for subgroup growth of Chevalley groups over $\mathbf{G}(\mathbb{F}_p[[t]])$. We introduce a new parameter, the ridgeline number $v(\mathbf{G})$, and give new bounds for the subgroup growth…
Fixing a subgroup $\Gamma$ in a group $G$, the full commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ of $G$ with $[\Gamma: \Gamma \cap \Delta][\Delta : \Gamma \cap \Delta] \leq n$. For…
The residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ of a finitely generated group $G$ is a function that gives the smallest value of the index $[G:N]$ with $N$ a normal subgroup not containing a non-trivial element $g$,…
Given a finitely generated residually finite group $G$, the residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ bounds the size of a finite group $Q$ needed to detect an element of norm at most $r$. More specifically, if…
Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the…
We prove a quantitative refinement of the statement that groups of polynomial growth are finitely presented. Let $G$ be a group with finite generating set $S$ and let $\operatorname{Gr}(r)$ be the volume of the ball of radius $r$ in the…
Let G be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if G has the congruence subgroup property, then the number of n-dimensional irreducible representations of G grows like n^a, where a is a…
In this article we investigate the L^1-norm of certain functions on groups called divisibility functions. Using these functions, their connection to residual finiteness, and integration theory on profinite groups, we define the residual…
In [K. Bou-Rabee, B. Seward, J. Reine Angwe. Math. 2016] Bou-Rabee and Seward constructed examples of finitely generated residually finite groups $G$ whose residual finiteness growth function $\mathcal{F}_G$ can be at least as fast as any…
We study a function $\mathcal{L}_{\Gamma}$ which quantifies the LEF (local embeddability into finite groups) property for a finitely generated group $\Gamma$. We compute this "LEF growth" function in some examples, including certain wreath…