arXiv subjects

Emmanuel Rauzy

Publications and source records attributed to Emmanuel Rauzy.

12 recordsLinked to original sources

arXiv2026

Residual properties of finitely generated groups in the Weihrauch lattice

Consider, on the space of marked groups, the map $\mathrm{Res}_{\mathcal{C}}$ which associates to a marked group its greatest residually-$\mathcal{C}$ quotient, for different sets $\mathcal{C}$ of groups. Except for trivial cases, this map is discontinuous. We use the Weihrauch lattice to quantify how discontinuous it is. We show that equational noetherianity of $\mathcal{C}$ and whether the set of residually-$\mathcal{C}$ groups is a quasivariety both can be characterized in terms of the position of $\mathrm{Res}_{\mathcal{C}}$ within the Weihrauch lattice. We give exact classifications of $\mathrm{Res}_{\mathcal{C}}$, for $\mathcal{C}$ one of: the set of finite groups, of nilpotent groups, of $k$-nilpotent groups, $k\ge1$, of finitely presentable groups, of LEF groups, of torsion free groups.

math.GR↗

arXiv2025

Computable Bases

In computable analysis typically topological spaces with countable bases are considered. The Theorem of Kreitz-Weihrauch implies that the subbase representation of a second-countable $T_0$ space is admissible with respect to the topology that the subbase generates. We consider generalizations of this setting to bases that are representable, but not necessarily countable. We introduce the notions of a computable presubbase and a computable prebase. We prove a generalization of the Theorem of Kreitz-Weihrauch for the presubbase representation that shows that any such representation is admissible with respect to the topology generated by compact intersections of the presubbase elements. For computable prebases we obtain representations that are admissible with respect to the topology that they generate. These concepts provide a natural way to investigate many topological spaces that have been studied in computable analysis. The benefit of this approach is that topologies can be described by their usual subbases and standard constructions for such subbases can be applied. Finally we discuss a Galois connection between presubbases and representations of $T_0$ spaces that indicates that presubbases and representations offer particular views on the same mathematical structure from different perspectives.

math.LO↗

arXiv2025

Effective bases and notions of effective second countability in computable analysis

We investigate different notions of "computable topological base" for represented spaces. We show that several non-equivalent notions of bases become equivalent when we consider computably enumerable bases. This indicates the existence of a robust notion of computably second countable represented space. These spaces are precisely those introduced by Grubba and Weihrauch under the name "computable topological spaces". The present work thus clarifies the articulation between Schr\"oder's approach to computable topology based on the Sierpinski representation and other approaches based on notions of computable bases. These other approaches turn out to be compatible with the Sierpinski representation approach, but also strictly less general. We revisit Schr\"oder's Effective Metrization Theorem, by showing that it characterizes those represented spaces that embed into computable metric spaces: those are the computably second countable strongly computably regular represented spaces. Finally, we study different forms of open choice problems. We show that having a computable open choice is equivalent to being computably separable, but that the "non-total open choice problem", i.e., open choice restricted to open sets that have non-empty complement, interacts with effective second countability in a satisfying way.

math.LO↗

arXiv2025

A robust family of residually finite groups; spectra of residual finiteness growth, computability properties, and other applications (with an appendix by Arman Darbinyan and Emmanuel Rauzy)

In this paper, we introduce a family of residually finite groups that helps us to systematically study the residual finiteness growth function (RFG) from various perspectives. First, by strengthening results of Bou-Rabee and Seward and also of Bradford, we show that any non-decreasing function $f: \nn \rightarrow \nn$ that satisfies $f(n) > \exp{(\varepsilon n\log{n})}$ for some $\varepsilon>0$ can be realized (up to the standard equivalence) as RFG function of a two-generated residually finite group. Moreover, such a group can be found among solvable groups of derived length $3$; due to what, in a strengthened way, we extend a theorem of Kharlampovich, Miasnikov and Sapir. {Next, we consider computability aspects related to those growth functions. In particular, we characterize the decidability of the word problem in residually finite groups with respect to \emph{individual residual finiteness depth} functions. Then, we give a full description of sufficiently fast {growing} functions that are realizable as RFG for some group \emph{with decidable word problem} in terms of \emph{left-computable functions.} We also show that a Turing degree can be realized via RFG of a group with decidable word problem if and only if it is recursively enumerable. Finally, applying the introduced theoretical framework, we answer several open questions and extend known results. For example, answering a question of Minasyan, by providing a construction, we show the existence of conjugacy separable groups with decidable word problem and undecidable conjugacy problem. Two more applications, including an answer to a question by Nies, can be found in the appendix coauthored with Rauzy.}

math.GR↗

arXiv2024

Groups with presentations in EDT0L

To any family of languages LAN, let us associate the class, denoted $\pi(\text{LAN})$, of finitely generated groups that admit a group presentation whose set of relators forms a language in LAN. We show that the class of L-presented groups, as introduced by the first author in 2003, is exactly the class of groups that admit presentations in the family of languages EDT0L. We show that the marked isomorphism problem is not semi-decidable for groups given by EDT0L presentations, contrary to the finite presentation case. We then extend and unify results of the first author with Eick and Hartung about nilpotent and finite quotients, by showing that it is possible to compute the marked hyperbolic and marked metabelian quotients of a group given by an EDT0L presentation. Finally, we show how the results about quotient computations enable the construction of recursively presented groups that do not have EDT0L presentations, thus proving $\pi(\text{EDT0L})\ne \pi(\text{REC})$. This is done by building a residually nilpotent group with solvable word problem whose sequence of maximal nilpotent quotients is non-computable.

math.GR↗

arXiv2023

A generalization of Markov's approach to the continuity problem for Type 1 computable functions

We axiomatize and generalize Markov's approach to the continuity problem for Type 1 computable functions, i.e. the problem of finding sufficient conditions on a computable topological space to obtain a theorem of the form "computable functions are (effectively) continuous". We introduce different notions of effective closure. These notions of effective closure lead to different notions of effective discontinuity at a point. We give conditions that prevent computable functions from having effective discontinuities. We finally show that results that forbid effective discontinuities can be turned into (abstract) continuity results on spaces where the closure and effective closure of semi-decidable sets naturally coincide. This happens for instance on spaces which admit a dense and computable sequence.

math.LO↗

arXiv2023

New definitions in the theory of Type 1 computable topological spaces

In 1957, Lacombe initiated a systematic study of the different possible notions of "computable topological spaces". However, he interrupted this line of research, settling for the idea that "computably open sets should be computable unions of basic open sets". We explain the limits of this approach, which in particular is not general enough to account for all spaces that admit a computable metric. We give a general notion of Type 1 computable topological space that does not rely on a notion of effective basis. Building on the work of Spreen, we show that the use of a $\textit{formal inclusion relation}$ should be systematized. We give the first general definition of the computable topology associated to a computable metric that does not rely on effective separability. This definition can be translated to other constructive settings, and its relevance goes beyond that of Type 1 computability. Finally, we give a new version of a theorem of Moschovakis, by showing that for an appropriate notion of effective basis, the "computably open sets" reduce to "computable unions of basic open sets" on computably separable spaces.

math.LO↗

arXiv2023

Multi-representation associated to the numbering of a subbasis and formal inclusion relations

We revisit Dieter Spreen's notion of a representation associated to a numbered basis equipped with a strong inclusion relation. We show that by relaxing his requirements, we obtain different classically considered representations as subcases, including representations considered by Grubba, Weihrauch and Schr\"oder. We show that the use of an appropriate strong inclusion relation guarantees that the representation associated to a computable metric space seen as a topological space always coincides with the Cauchy representation. We also show how the use of a formal inclusion relation guarantees that when defining multi-representations on a set and on one of its subsets, the obtained multi-representations will be compatible, i.e. inclusion will be a computable map. The proposed definitions are also more robust under change of equivalent bases.

math.LO↗

arXiv2021

Computable analysis on the space of marked groups

We begin the systematic study of decision problems for finitely generated groups given by a solution to their word problem. We relate this to the study of computable analysis on the space of marked groups. We point out that several distinct approaches to computable analysis, some of which are sometimes considered obsolete, yield relevant results. In particular, we give necessary and sufficient conditions in terms of Banach-Mazur computability for the existence of a finitely presented group with solvable word problem but whose subgroups with a certain property cannot be recognized. We classify group properties in different effective Borel hierarchies. For most common group properties, the classical and effective Borel classifications coincide. However, we show that the set of LEF groups is a closed set that is computably a $G_{\delta}$, but not computably closed. Finally, we show that the space of marked groups is a Polish space which is not $\textit{computably Polish}$, because it does not admit a dense and computable sequence. This poses several interesting problems in terms of computable topology. The space of marked groups is the first natural example of this kind.

math.GR↗

arXiv2021

Remarks and problems about algorithmic descriptions of groups

Motivated by a theorem of Groves and Wilton, we propose the study of the lattice of numberings of isomorphism classes of marked groups as a rigorous and comprehensive framework to study global decision problems for finitely generated groups. We establish the Rice and Rice-Shapiro Theorems for recursive presentations, and establish similar results for co-recursive presentations. We give an algorithmic characterization of finitely presentable groups in terms of semi-decidability of two decision problems: the word problem and the marked quotient problem, which we introduce. We explain how this result can be used to define algorithmic generalizations of finite presentations. Finally, we discuss how the Adian-Rabin Theorem provides incomplete answers in several respects.

math.GR↗

arXiv2021

Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem

We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite group with solvable word problem for which there is no effective method that allows, given some non-identity element, to find a morphism onto a finite group in which this element has a non-trivial image. We also prove that the depth function of this group grows faster than any recursive function.

math.GR↗

arXiv2020

Computability of finite quotients of finitely generated groups

We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can be even not recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem while still not having computable finite quotients.

math.GR↗