Group Theory

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Showing new listings for Thursday, 21 November 2024

New submissions (showing 1 of 1 entries)

[1] arXiv:2411.12848 [pdf, html, other]

Title: On the abscissae of Weil representation zeta functions for procyclic groups

Steffen Kionke

Comments: 9 pages, comments welcome

Subjects: Group Theory (math.GR); Number Theory (math.NT)

A famous conjecture of Chowla on the least primes in arithmetic progressions implies that the abscissa of convergence of the Weil representation zeta function for a procyclic group $G$ only depends on the set $S$ of primes dividing the order of $G$ and that it agrees with the abscissa of the Dedekind zeta function of $\mathbb{Z}[p^{-1}\mid p \not\in S]$. Here we show that these consequences hold unconditionally for random procyclic groups in a suitable model. As a corollary, every real number $1 \leq \beta \leq 2$ is the Weil abscissa of some procyclic group.

Cross submissions (showing 4 of 4 entries)

[2] arXiv:2411.12881 (cross-list from math.GT) [pdf, html, other]

Title: Loops, Holonomy and Signature

Juan Alonso, Juan Manuel Burgos, Miguel Paternain

Comments: 11 pages, no figures

Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Group Theory (math.GR)

We show that there is a topology on the group of loops in euclidean space such that this group is embedded in a Lie group which is simple relative to the loops. An extension of this Lie group gives the structural group of a principal bundle with connection whose holonomy coincides with the Chen signature map. We also give an alternative geometric new proof of Chen signature Theorem.

[3] arXiv:2411.12927 (cross-list from math.GT) [pdf, html, other]

Title: Classification of Stable Surfaces with respect to Automatic Continuity

Mladen Bestvina, George Domat, Kasra Rafi

Comments: 37 pages, 5 figures

Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We provide a complete classification of when the homeomorphism group of a stable surface, $\Sigma$, has the automatic continuity property: Any homomorphism from Homeo$(\Sigma)$ to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of $\Sigma$ has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. Applying this framework, we also show that the homeomorphism group of any stable second countable Stone space has the automatic continuity property. Under the presence of stability this answers two questions of Mann.

[4] arXiv:2411.13272 (cross-list from math.AT) [pdf, html, other]

Title: Geometric invariants of locally compact groups: the homological perspective

Kai-Uwe Bux, Elisa Hartmann, José Pedro Quintanilha

Subjects: Algebraic Topology (math.AT); Group Theory (math.GR); Metric Geometry (math.MG)

In this paper we develop the homological version of $\Sigma$-theory for locally compact Hausdorff groups, leaving the homotopical version for another paper. Both versions are connected by a Hurewicz-like theorem. They can be thought of as directional versions of type $\mathrm{CP}_m$ and type $\mathrm{C}_m$, respectively. And classical $\Sigma$-theory is recovered if we equip an abstract group with the discrete topology. This paper provides criteria for type $\mathrm{CP}_m$ and homological locally compact $\Sigma^m$. Given a short exact sequence with kernel of type $\mathrm{CP}_m$, we can derive $\Sigma^m$ of the extension on the sphere that vanishes on the kernel from the quotient and likewise. Given a short exact sequence with abelian quotient, $\Sigma$-theory on the extension can tell if the kernel is of type $\mathrm{CP}_m$.

[5] arXiv:2411.13294 (cross-list from math.MG) [pdf, html, other]

Title: Topological expanders, coarse geometry and thick embeddings of complexes

David Hume

Comments: 47 pages, 2 figures

Subjects: Metric Geometry (math.MG); Combinatorics (math.CO); Group Theory (math.GR); Geometric Topology (math.GT)

We quantify the topological expansion properties of bounded degree simplicial complexes in terms of a family of sublinear functions, in analogy with the separation profile of Benjamini-Schramm-Timár for classical expansion of bounded degree graphs. We prove that, like the separation profile, these new invariants are monotone under regular maps between complexes satisfying appropriate higher connectivity assumptions. In the dimension $1$ case, we recover the cutwidth profile of Huang-Hume-Kelly-Lam. We also prove the seemingly new result that any $1$-dimensional topological expander necessarily contains a graphical expander.
In higher dimensions, we give full calculations of these new invariants for Euclidean spaces, which are natural analogues of waist and width-volume inequalities due to Gromov and Guth respectively. We present several other methods of obtaining upper bounds including naïve (yet useful) direct product and fibring theorems, and show how lower bounds can be obtained via thick embeddings of complexes, in analogy with previous work of Barrett-Hume. Using this, we find lower bounds for $k$-expansion of $(k+1)$-fold horocyclic products of trees, and for rank $k$ symmetric spaces of non-compact type. As a further application, we prove that for every $k\geq 2$ there is no coarse embedding (and more generally, no regular map) from the $k$-fold horocyclic product of $3$-regular trees to either any product $(\mathbb{H}^2)^{k-2}\times H \times D$ where $\mathbb{H}^2$ is the real hyperbolic plane, $H$ is a bounded degree hyperbolic graph and $D$ is a doubling metric space, or to any symmetric space whose non-compact factor has corank (dimension minus rank) is strictly less than $k$.

Replacement submissions (showing 4 of 4 entries)

[6] arXiv:2312.05427 (replaced) [pdf, html, other]

Title: Liftable self-similar groups and scale groups

Rostislav Grigorchuk, Dmytro Savchuk

Comments: 50 pages, 9 figures

Subjects: Group Theory (math.GR)

We canonically identify the groups of isometries and dilations of local fields and their rings of integers with subgroups of the automorphism group of the $(d+1)$-regular tree $\widetilde T_{d+1}$, where $d$ is the residual degree. Then we introduce the class of liftable self-similar groups acting on a $d$-regular rooted tree whose ascending HNN extensions act faithfully and vertex transitively on $\widetilde T_{d+1}$ fixing one of the ends. The closures of these extensions in $\mathrm{Aut}(\widetilde T_{d+1})$ are totally disconnected locally compact group that belong to the class of scale groups. We give numerous examples of liftable groups coming from self-similar groups acting essentially freely or groups admitting finite $L$-presentations. In particular, we show that the finitely presented group constructed by the first author and the finitely presented HNN extension of the Basilica group embed into the group $\mathcal D(\mathbb Q_2)$ of dilations of the field $\mathbb Q_2$ of $2$-adic numbers. These actions, translated to $\widetilde T_3$, are 2-transitive on the punctured boundary of $\widetilde T_3$. Also we explore scale-invariant groups with the purpose of getting new examples of scale groups.

[7] arXiv:2401.00404 (replaced) [pdf, other]

Title: Explicit Generators for the Stabilizers of Rational Points in Thompson's Group $F$

Krystofer Baker, Dmytro Savchuk

Comments: 19 pages, 9 figures and pictures

Subjects: Group Theory (math.GR)

We construct explicit finite generating sets for the stabilizers in Thompson's group $F$ of rational points of a unit interval or a Cantor set. Our technique is based on the Reidemeister-Schreier procedure in the context of Schreier graphs of such stabilizers in $F$. It is well known that the stabilizers of dyadic rational points are isomorphic to $F\times F$ and can thus be generated by 4 explicit elements. We show that the stabilizer of every non-dyadic rational point $b\in (0,1)$ is generated by 5 elements that are explicitly calculated as words in generators $x_0, x_1$ of $F$ that depend on the binary expansion of $b$. We also provide an alternative simple proof that the stabilizers of all rational points are finitely presented.

[8] arXiv:2410.20412 (replaced) [pdf, html, other]

Title: Geodesic languages for rational subsets and conjugates in virtually free groups

André Carvalho, Pedro V. Silva

Comments: 16 pages, comments are welcome

Subjects: Group Theory (math.GR); Formal Languages and Automata Theory (cs.FL)

We prove that a subset of a virtually free group is rational if and only if the language of geodesic words representing its elements (in any generating set) is rational and that the language of geodesics representing conjugates of elements in a rational subset of a virtually free group is context-free. As a corollary, the doubly generalized conjugacy problem is decidable for rational subsets of finitely generated virtually free groups: there is an algorithm taking as input two rational subsets $K_1$ and $K_2$ of a virtually free group that decides whether there is one element of $K_1$ conjugate to an element of $K_2$. For free groups, we prove that the same problem is decidable with rational constraints on the set of conjugators.

[9] arXiv:2411.03789 (replaced) [pdf, html, other]

Title: Lower bounds on the essential dimension of reductive groups

Danny Ofek

Comments: 51 pages; A few typos and an error in the computation of the lower bound on ed(E_7^{ad};2) were corrected (the correct lower bound is 19)

Subjects: Group Theory (math.GR); Algebraic Geometry (math.AG)

We introduce a new technique for proving lower bounds on the essential dimension of split reductive groups. As an application, we strengthen the best previously known lower bounds for various split simple algebraic groups, most notably for the exceptional group $E_8$. In the case of the projective linear group $\operatorname{PGL}_n$, we recover A. Merkurjev's celebrated lower bound with a simplified proof. Our technique relies on decompositions of loop torsors over valued fields due to P. Gille and A. Pianzola.