On Idempotent Measures of Small Norm

<p>In this Master’s Thesis, we set up the groundwork for [8], a paper co-written by the author and Hung Pham. We summarise the Fourier and Fourier-Stieltjes algebras on both abelian and general locally compact groups. Let Г be a locally compact group. We answer two questions left open in [11] and [13]: 1. When Г is abelian, we prove that if ϰs ∈ B(Г) is an idempotent with norm 1 < ||ϰs|| < 4/3 then S is the union of two cosets of an open subgroup of Г. 2. For general Г, we prove that if ϰs ∈ McbA(Г) is an idempotent with norm ||ϰs||cb < 1+√2/2 , then S is an open coset in Г.</p>

On Idempotent Measures of Small Norm

<p>In this Master’s Thesis, we set up the groundwork for [8], a paper co-written by the author and Hung Pham. We summarise the Fourier and Fourier-Stieltjes algebras on both abelian and general locally compact groups. Let Г be a locally compact group. We answer two questions left open in [11] and [13]: 1. When Г is abelian, we prove that if ϰs ∈ B(Г) is an idempotent with norm 1 < ||ϰs|| < 4/3 then S is the union of two cosets of an open subgroup of Г. 2. For general Г, we prove that if ϰs ∈ McbA(Г) is an idempotent with norm ||ϰs||cb < 1+√2/2 , then S is an open coset in Г.</p>

Compact groups with many elements of bounded order

Lévai and Pyber proposed the following as a conjecture: Let G be a profinite group such that the set of solutions of the equation {x^{n}=1} has positive Haar measure. Then G has an open subgroup H and an element t such that all elements of the coset tH have order dividing n (see [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences, Novosibirisk, 2019; Problem 14.53]). The validity of the conjecture has been proved in [L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1–7] for {n=2}. Here we study the conjecture for compact groups G which are not necessarily profinite and {n=3}; we show that in the latter case the group G contains an open normal 2-Engel subgroup.

The virtually generating graph of a profinite group

We consider the graph $\Gamma _{\text {virt}}(G)$ whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph $\Delta _{\text {virt}}(G)$ obtained from $\Gamma _{\text {virt}}(G)$ by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that $\Delta _{\operatorname {\mathrm {virt}}}(G)$ has precisely t connected components. Moreover, we study the graph $\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$ , whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph $\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.

Characterization of non-compact locally compact groups by cocompact subgroups

A subgroup H of a topological group G is called cocompact (or uniform) if the quotient space {G/\overline{H}} is compact, where {\overline{H}} denotes the closure of H in G. The purpose of this paper is to give a characterization of non-compact locally compact groups with the property that every non-trivial closed (respectively, open) subgroup is cocompact (respectively, has finite index).

Large normal subgroup growth and large characteristic subgroup growth

The fastest 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 Γ be a group and Δ a subgroup of finite index. Suppose Δ has normal subgroup growth of type {n^{\log n}}. Does Γ have normal subgroup growth of type {n^{\log n}}? We give a positive answer in some cases, generalizing a result of Müller and the second author and a result of Gerdau. For instance, suppose G is a profinite group and H an open subgroup of G. We show that if H is a generalized Golod–Shafarevich group, then G has normal subgroup growth of type {n^{\log n}}. We also use our methods to show that one can find a group with characteristic subgroup growth of type {n^{\log n}}.

Bounding the covolume of lattices in products

We study lattices in a product $G=G_{1}\times \cdots \times G_{n}$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_{i}$ is non-compact and every closed normal subgroup of $G_{i}$ is discrete or cocompact (e.g. $G_{i}$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $\unicode[STIX]{x1D6E4}$ with dense projections is finite. The same result holds if $\unicode[STIX]{x1D6E4}$ is non-uniform, provided $G$ has Kazhdan’s property (T). We show that for any compact subset $K\subset G$, the collection of discrete subgroups $\unicode[STIX]{x1D6E4}\leqslant G$ with $G=\unicode[STIX]{x1D6E4}K$ and dense projections is uniformly discrete and hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $\operatorname{Aut}(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.

DISTALITY OF CERTAIN ACTIONS ON -ADIC SPHERES

Consider the action of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on the $p$-adic unit sphere ${\mathcal{S}}_{n}$ arising from the linear action on $\mathbb{Q}_{p}^{n}\setminus \{0\}$. We show that for the action of a semigroup $\mathfrak{S}$ of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on ${\mathcal{S}}_{n}$, the following are equivalent: (1) $\mathfrak{S}$ acts distally on ${\mathcal{S}}_{n}$; (2) the closure of the image of $\mathfrak{S}$ in $\operatorname{PGL}(n,\mathbb{Q}_{p})$ is a compact group. On ${\mathcal{S}}_{n}$, we consider the ‘affine’ maps $\overline{T}_{a}$ corresponding to $T$ in $\operatorname{GL}(n,\mathbb{Q}_{p})$ and a nonzero $a$ in $\mathbb{Q}_{p}^{n}$ satisfying $\Vert T^{-1}(a)\Vert _{p}<1$. We show that there exists a compact open subgroup $V$, which depends on $T$, such that $\overline{T}_{a}$ is distal for every nonzero $a\in V$ if and only if $T$ acts distally on ${\mathcal{S}}_{n}$. The dynamics of ‘affine’ maps on $p$-adic unit spheres is quite different from that on the real unit spheres.

Homomorphisms into totally disconnected, locally compact groups with dense image

Let {\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair {(G,\phi^{-1}(L))} , where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.