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• ORCID: 0000-0003-0228-2585
• arXiv: loh_c_1
• mathscinet: 780073 (subscription required)
• zbMATH Open: loh.clara
• DFG GEPRIS: 179678034
• math genealogy project: 118955

books

• Exploring Formalisation. A Primer in Human-Readable Mathematics in Lean 3 with Examples from Simplicial Topology
  Homepage of this book project (including Lean src),
  Surveys and Tutorials in the Applied Mathematical Sciences , volume 11, Springer, 09/2022.
  print ISBN 978-3-031-14648-0, ebook ISBN 978-3-031-14649-7, DOI 10.1007/978-3-031-14649-7
  Please note that due to system changes the URLs referring to my homepage in the book are not working anymore; the new prefix is https://loeh.app.ur.de
  
  
• Ergodic theoretic methods in group homology. A minicourse on L^2-Betti numbers in group theory. ( draft pdf ).
  SpringerBriefs in Mathematics, Springer, ISBN 978-3-030-44219-4, DOI 10.1007/978-3-030-44220-0, 03/2020.
  Errata: pdf
  
  
• Geometric Group Theory. An Introduction. ( draft pdf ).
  Universitext, Springer, ISBN 978-3-319-72253-5, 01/2018.
  Errata: pdf
  
  
• Quod erat knobelandum, Themen, Aufgaben und Lösungen des Schülerzirkels Mathematik der Universität Regensburg.
  Editors: Clara Löh, Stefan Krauss, Niki Kilbertus
  Authours of the second edition: Karin Binder, Georg Bruckmaier, Andreas Eberl, Alexander Engel, Gerrit, Herrmann, Timo Keller, Niki Kilbertus, Stefan Krauss, Clara Löh, Christian Nerf, Theresa Stoiber, Jan-Hendrik Treude, Alexander Voitovitch
  Springer Spektrum, ISBN 978-3-662-58724-9, 03/2019.

Authours of the first edition: Andreas Eberl, Alexander Engel, Timo Keller, Niki Kilbertus, Stefan Krauss, Clara Löh, Christian Nerf, Theresa Stoiber, Jan-Hendrik Treude, Alexander Voitovitch
Springer Spektrum, ISBN 978-3-662-48955-0, 04/2016.

journal publications

• Bounded acylicity and relative simplicial volume ( pdf ).
  Joint work with Kevin Li, Marco Moraschini.
  Also available at the arXiv: 2202.05606 [math.AT]
  To appear in Journal of Topology and Analysis
  
  
• Universal finite functorial semi-norms ( pdf ).
  Joint work with Johannes Witzig
  Also available at the arXiv: 2209.12971 [math.CT]
  Original publication in Applied Categorical Structures: Appl. Categor. Struct., 32, paper 19, 2024. The DOI is 10.1007/s10485-024-09777-w
  
  
• Integral foliated simplicial volume and ergodic decomposition ( pdf ).
  Joint work with Giovanni Sartori
  Also available at the arXiv: 2211.00337 [math.GT]
  To appear in Annales Mathématiques Blaise Pascal.
  
  
• Exponential growth rates in hyperbolic groups [after Fujiwara and Sela] ( pdf ).
  Exposée 1206 for the Séminaire Bourbaki, 04/2023.
  Also available at the arXiv: 2304.04424 [math.GR]
  Original publication in Astérisque, 446, Séminaire Bourbaki, vol. 2022/2023, Exposée 1206, 365--382, 2023. The DOI is 10.24033/ast.1215
  
  
• Median quasimorphisms on CAT(0) cube complexes and their cup products ( pdf ).
  Joint work with Benjamin Brück, Francesco Fournier Facio
  Also available at the arXiv: 2209.05811 [math.GR]
  Original publication in Geometriae Dedicata: Geom Dedicata 218:28, 2024. The DOI is 10.1007/s10711-023-00870-3
  
  
• L^2-Betti numbers and computability of reals ( pdf ).
  Joint work with Matthias Uschold
  (Partial) implementation in Lean
  Also available at the arXiv: 2202.03159 [math.GR]
  Original publication in Computability: Computability, 12(2), pp. 175--201, 2023. The DOI is 10.3233/COM-220416.
  
  
• The spectrum of simplicial volume with fixed fundamental group ( pdf ).
  Also available at the arXiv: 2205.14877 [math.GT]
  Original publication in Geometriae Dedicata: Geom Dedicata 217, article 16, 2023. The DOI is 10.1007/s10711-022-00749-9. Published pdf: pdf
  
  
• Bounded cohomology of finitely presented groups: vanishing, non-vanishing, and computability ( pdf ).
  Joint work with Francesco Fournier Facio, Marco Moraschini.
  Also available at the arXiv: 2106.13567 [math.GR]
  Original publication in Annali della Scuola Normale Superiore di Pisa: Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. XXIV, 1169--1202, 2024. The DOI is 10.2422/2036-2145.202201_003
  
  
• On the simplicial volume and the Euler characteristic of (aspherical) manifolds ( pdf ).
  Joint work with Marco Moraschini, George Raptis.
  Also available at the arXiv: 2109.08115 [math.AT]
  Original publication in Research in the Mathematical Sciences: Res Math Sci, 9, article 44, 2022. The DOI is 10.1007/s40687-022-00341-3. Published pdf: pdf
  
  
• Amenable covers and integral foliated simplicial volume ( pdf ).
  Joint work with Marco Moraschini, Roman Sauer.
  Also available at the arXiv: 2112.12223 [math.GT]
  Original publication in New York Journal of Mathematics: New York J. Math., 28, pp. 1112--1136, 2022. Published pdf: pdf
  
  
• Bounded cohomology and binate groups ( pdf ).
  Joint work with Francesco Fournier Facio, Marco Moraschini.
  Also available at the arXiv: 2111.04305 [math.GR]
  Original publication in Journal of the Australian Mathematical Society. The DOI is 10.1017/S1446788722000106: J. Aust. Math. Soc., 115(2), pp. 204--239, 2023. Published pdf: pdf
  
  
• Topological volumes of fibrations: A note on open covers ( pdf ).
  Joint work with Marco Moraschini.
  Also available at the arXiv: 2104.06038 [math.GT]
  Original publication in Proceedings A of the Royal Society of Edinburgh : Proc. Roy. Soc. Edinburgh Sect. A, 152, pp. 1340--1360, 2022. The DOI is 10.1017/prm.2021.64
  
  
• The spectrum of simplicial volume of non-compact manifolds ( pdf ).
  Joint work with Nicolaus Heuer
  Also available at the arXiv: 2010.12945 [math.GT]
  Original publication in Geometriae Dedicata: Geom. Dedicata, 215, pp. 243--253, 2021. The DOI is 10.1007/s10711-021-00647-6
  
  
• Simplicial volume of one-relator groups and stable commutator length ( pdf ).
  Joint work with Nicolaus Heuer
  Implementation of lallop in MATLAB
  Implementation of lallop in Haskell
  Also available at the arXiv: 1911.02470 [math.GT]
  Original publication in Algebraic and Geometric Topology: Alg. Geom. Top., 22(4), pp. 1615--1661, 2022. The DOI is 10.2140/agt.2022.22.1615.
  
  
• Stable integral simplicial volume of 3-manifolds ( pdf ).
  Joint work with Daniel Fauser, Marco Moraschini, José Pedro Quintanilha
  Also available at the arXiv: 1910.06120 [math.GT]
  Original publication in Journal of Topology: J. Topol., 14(2), pp. 608--640, 2021. The DOI is 10.1112/topo.12193.
  
  
• Epimorphism testing with virtually Abelian targets ( pdf ).
  Joint work with Stefan Friedl
  Also available at the arXiv: 2010.07537 [math.GR]
  Original publication in Confluentes Mathematici: Confluentes Math., 13(1), pp. 61--78, 2021. published pdf
  
  
• Amenable category and complexity ( pdf ).
  Joint work with Pietro Capovilla and Marco Moraschini
  Also available at the arXiv: 2012.00612 [math.AT]
  Original publication in Algebraic and Geometric Topology: Alg. Geom. Topol., 22(3), pp. 1417--1459, 2022. The DOI is 10.2140/agt.2022.22.1417.
  
  
• Transcendental simplicial volumes ( pdf ).
  Joint work with Nicolaus Heuer
  Also available at the arXiv: 1911.06386 [math.GT]
  Original publication in Annales de l'Institut Fourier: Ann. Inst. Fourier, 74(2), pp. 763--781, 2024. The DOI is 10.5802/aif.3597
  
  
• The spectrum of simplicial volume ( pdf ).
  Joint work with Nicolaus Heuer
  Also available at the arXiv: 1904.04539 [math.GT]
  Original publication in Inventiones mathematicae: Invent. math., 223, pp. 103--148, 2021. The DOI is 10.1007/s00222-020-00989-0
  
  
• A note on p-adic simplicial volumes ( pdf ).
  Joint work with Steffen Kionke
  Also available at the arXiv: 2003.10756 [math.GT]
  Original publication in Glasgow Mathematical Journal: Glasgow Math. J., 63(3), pp. 563--583, 2021. The DOI is 10.1017/S0017089520000385
  
  
• Bounded cohomology of amenable covers via classifying spaces ( pdf ).
  Joint work with Roman Sauer
  Also available at the arXiv: 1910.11716 [math.AT]
  Original publication in L'Enseignement Mathématique: Enseign. Math., 66(1/2), pp. 151--172, 2020. The DOI is 10.4171/LEM/66-1/2-8
  
  
• Simplicial volume via normalised cycles ( pdf ).
  Joint work with Marco Moraschini
  Also available at the arXiv: 2003.02584 [math.AT]
  Original publication in Proceedings of the AMS: Proc. Amer. Math. Soc., 148(12), pages 5437--5440, 2020. The DOI is 10.1090/proc/15201
  
  
• Polynomially weighted l^p-completions and group homology ( pdf ).
  Joint work with Alexander Engel
  Also available at the arXiv: 1903.11486 [math.GR]
  Original publication in Journal of Combinatorial Algebra: J. Comb. Algebra, volume 4(1), pp. 87--109, 2020. The DOI is 10.4171/JCA/40
  
  
• Cost vs. integral foliated simplicial volume ( pdf ).
  Also available at the arXiv: 1809.09660 [math.GT]
  Original publication in Groups, Geometry, and Dynamics: Groups Geom. Dyn., 14(3), pp. 899--916, 2020. The DOI is 10.4171/GGD/568
  
  
• Simplicial volume with F_p-coefficients ( pdf ).
  Also available at the arXiv: 1808.09497 [math.GT]
  Original publication in Periodica Mathematica Hungarica: Period Math Hung, volume 80(1), pp. 38--58, 2020. The DOI is DOI 10.1007/s10998-019-00298-x
  Full-text pdf: pdf
  
  
• Integral approximation of simplicial volume of graph manifolds ( pdf ).
  Joint work with Daniel Fauser and Stefan Friedl
  Also available at the arXiv: 1807.10522 [math.GT]
  Original publication in Bulletin of the London Mathematical Society: Bull. Lond. Math. Soc., 51(4), pp. 715--731, 2019. The DOI is DOI 10.1112/blms.12266
  
  
• Variations on the theme of the uniform boundary condition ( pdf ).
  Joint work with Daniel Fauser
  Also available at the arXiv: 1703.01108 [math.GT]
  Original publication in Journal of Topology and Analysis: J. Topol. Anal., 13(1), pp. 147--174, 2021. The DOI is DOI 10.1142/S1793525320500090
  
  
• Exotic finite functorial semi-norms on singular homology ( pdf ).
  Joint work with Daniel Fauser
  Also available at the arXiv: 1605.04093 [math.GT]
  Original publication in Glasgow Mathematical Journal: Glasgow Mathematical Journal, volume 61(2), pp. 287--295, 2019. The DOI is DOI 10.1017/S0017089518000216
  
  
• Odd manifolds of small integral simplicial volume ( pdf ).
  Also available at the arXiv: 1509.00204 [math.GT]
  Original publication in Arkiv för Matematik : Arkiv för Matematik, volume 56(2), pp. 351--375, 2018. The DOI is DOI 10.4310/ARKIV.2018.v56.n2.a10
  
  
• Rank gradient vs. stable integral simplicial volume ( pdf ).
  Also available at the arXiv: 1704.05222 [math.GT]
  Original publication in Periodica Mathematica Hungarica: The DOI is DOI 10.1007/s10998-017-0212-1, Period Math Hung, volume 76(1), pp. 88--94, 2018.
  Full-text view-only pdf: pdf
  
  
• The l^infty-semi-norm on uniformly finite homology ( pdf ).
  Joint work with Francesca Diana
  Version with longer appendix: pdf
  Also available at the arXiv: 1502.01177 [math.MG]
  Original publication in Forum Mathematicum: Forum Math., volume 29(6), pp. 1325--1336, 2017. The DOI is 10.1515/forum-2015-0258.
  
  
• Integral foliated simplicial volume of aspherical manifolds ( pdf ).
  Joint work with Roberto Frigerio, Cristina Pagliantini, Roman Sauer
  Also available at the arXiv: 1506.05567 [math.GT]
  Original publication in Israel Journal of Mathematics: (link to pdf, public access view-only pdf). Israel J. Math., volume 216(2), pp. 707--751, 2016.
  
  
• On stability of non-domination under taking products ( pdf ).
  Joint work with D. Kotschick, Christoforos Neofytidis
  Also available at the arXiv: 1507.01413 [math.GT]
  Published by Proceedings of the AMS, Volume 144, pp. 2705--2710, 2016. The DOI is 10.10960/proc/13043
  
  
• A note on bounded-cohomological dimension of discrete groups ( pdf ).
  Also available at the arXiv: 1504.05760 [math.GR]
  Published by Journal of the Mathematical Society of Japan, Volume 69(2), pp. 715--734, 2017.
  
  
• Finite functorial semi-norms and representability ( pdf ).
  Also available at the arXiv: 1404.6557 [math.AT]
  Published by International Mathematics Research Notices, Volume 2016(12), pp. 3616--3638, 2016. The DOI is 10.1093/imrn/rnv250. (pdf)
  
  
• Integral foliated simplicial volume of hyperbolic 3-manifolds ( pdf ).
  Joint work with Cristina Pagliantini
  Also available at the arXiv: 1403.4518 [math.GT]
  Original publication in Groups, Geometry, and Dynamics : (link to pdf). Groups Geom. Dyn., volume 10(3), pp. 825--865, 2016.
  
  
• Functorial semi-norms on singular homology and (in)flexible manifolds ( pdf ).
  Joint work with Diarmuid Crowley
  Also available at the arXiv: 1103.4139 [math.GT]
  Published by Algebraic & Geometric Topology , Volume 15(3), pp. 1453--1499, 2015. The DOI is 10.2140/agt.2015.15.1453.
  
  
• Which finitely generated Abelian groups admit isomorphic Cayley graphs? ( pdf ).
  Also available at the arXiv: 1202.5484 [math.GR]
  Published by Geometriae Dedicata, Volume 164, pp. 97--111, 2013. The original publication is available at www.springerlink.com. The DOI is 10.1007/s10711-012-9761-x.
  
  
• Groups not presentable by products ( pdf ).
  Joint work with D. Kotschick
  Also available at the arXiv: 1003.3853 [math.GR]
  First published in Groups, Geometry, and Dynamics ( pdf ), Volume 7(1), pp. 181--204, 2013.
  
  
• Simplicial volume of Hilbert modular varieties ( pdf ).
  Joint work with Roman Sauer.
  Also available at the arXiv: 0706.3904.
  Published by Commentarii Mathematici Helvetici ( pdf ), Comment. Math. Helv., Volume 84, pp. 457--470, 2009.
  
  
• Degree theorems and Lipschitz simplicial volume for non-positively curved manifolds of finite volume ( pdf ).
  Joint work with Roman Sauer.
  Also available at the arXiv: 0710.1635.
  Published by the Journal of Topology ( pdf ), Journal of Topology, Volume 2, pp. 193--225, 2009.
  Also available through the Journal of Topology Virtual Issue
  
  
• Fundamental classes not representable by products ( pdf ).
  Joint work with D. Kotschick
  Also available at the arXiv: 0806.4540 [math.GT].
  Published by the Journal of the London Mathematical Society ( pdf ), J. London Math. Soc. Volume 79, Number 3, pp. 545--561 2009.
  
  
• Isomorphisms in l^1-homology ( pdf, previous version ).
  Also available at the arXiv: math.AT/0612589.
  Münster Journal of Mathematics ( pdf ), Volume 1, pp. 237-266, 2008.
  
  
• Measure homology and singular homology are isometrically isomorphic (pdf ).
  Also available at the arXiv: math.AT/0504103
  Published by Mathematische Zeitschrift. The original publication is available at www.springerlink.com. The DOI is 10.1007/s00209-005-0905-7. The complete reference of the printed version is: Mathematische Zeitschrift, Volume 253, Number 1, pp. 197-218, 2006.

preprints

• (11/2024) Algorithms in 4-manifold topology
  Joint work with Stefan Bastl, Rhuaidi Burke, Rima Chatterjee, Subhankar Dey, Alison Durst, Stefan Friedl, Daniel Galvin, Alejandro García Rivas, Tobias Hirsch, Cara Hobohm, Chun-Sheng Hsueh, Marc Kegel, Frieda Kern, Shun Ming Samuel Lee, Naageswaran Manikandan, Léo Mousseau, Lars Munser, Mark Pencovitch, Patrick Perras, Mark Powell, José Pedro Quintanilha, Lisa Schambeck, David Suchodoll, Martin Tancer, Annika Thiele, Paula Truöl, Matthias Uschold, Simona Veselá, Melvin Weiß, Magdalina von Wunsch-Rolshoven
  Also available at the arXiv: 2411.08775 [math.GT]
  
  
• (09/2024) The algebraic cheap rebuilding property (pdf).
  Joint work with Kevin Li, Marco Moraschini, Roman Sauer, Matthias Uschold
  Also available at the arXiv: 2409.05774 [math.GR]
  
  
• (11/2023) A quadratic lower bound for the number of minimal geodesics (pdf).
  Joint work with Bernd Ammann
  Also available at the arXiv: 2311.01626 [math.DG]
  
  
• (01/2023) A comment on the structure of graded modules over graded principal ideal domains in the context of persistent homology ( pdf ).
  Also available at the arXiv: 2301.11756 [math.AC]
  
  
• (03/2019) Residually finite categories ( pdf ).
  Also available at the arXiv: 1903.11488 [math.CT]
  
  
• (08/2015) Cubical simplicial volume of 3-manifolds ( pdf ).
  Joint work with Cristina Pagliantini, Sebastian Waeber
  Also available at the arXiv: 1508.03017 [math.GT]
  
  
• (09/2013) Which finitely generated Abelian groups admit equal growth functions? ( pdf ).
  Joint work with Matthias Mann
  Also available at the arXiv: 1309.3381 [math.GR]
  
  
• (04/2008) Fundamental classes of negatively curved manifolds cannot be represented by products of manifolds ( pdf ).
  Also available at the arXiv: 0804.3242v2 [math.AT]
  [Superseded by 0806.4540 [math.GT] above]

other contributions/notes

• Simplicial volume, survey page at the Manifold Atlas Project.
  Published also in the Bulletin of the Manifold Atlas (pdf),
  Bull. Man. Atl., pp. 7--18, 2011.
  
  
• Cubical simplicial volume of surfaces ( pdf ), note (11/2014; updated 02/2015, 04/2016).
  Joint with Christian Plankl
  Bull. Man. Atl., 2017.
  
  
• Explicit l^1-efficient cycles and amenable normal subgroups ( pdf ), expository note without original results (04/2017)
  Also available at the arXiv: 1704.05345 [math.GT]
  
  
• A roadmap to the (vanishing of the) Euler characteristic ( paper/poster pdf ).
  Joint work with George Raptis
  Also available at the arXiv: 2306.16933 [math.GT]
  
  
• List of all my typed lecture notes

implementation/formalisation of research work

• Lower bounds for amenable multiplicity
  (Partial) implementation in Lean of results from DOI 10.4171/LEM/66-1/2-8 / arXiv:1910.11716 [math.AT] and arXiv:2202.05606 [math.AT]
  
  
• Finite functorial semi-norms and representability
  (Partial) implementation in Lean of results from DOI 10.1093/imrn/rnv250 / arXiv:1404.6557 [math.AT]
  
  
• L^2-Betti numbers and computability of reals
  Joint work with Matthias Uschold
  (Partial) implementation in Lean of results from 2202.03159 [math.GR]
  
  
• lallop
  Joint work with Nicolaus Heuer
  Implementation of lallop in MATLAB
  Implementation of lallop in Haskell
  This is related to 1911.02470 [math.GT]
  

• Implementation/formalisation of teaching material is listed here.

doctoral thesis: l^1-Homology and Simplicial Volume

I wrote my doctoral thesis under supervision of Prof. Dr. W. Lück.

Abstract. Taking the l^1-completion and the topological dual of the singular chain complex gives rise to l^1-homology and bounded cohomology respectively. Unlike l^1-homology, bounded cohomology is quite well understood by the work of Gromov and Ivanov. We derive a mechanism linking isomorphisms on the level of homology of Banach chain complexes to isomorphisms on the level of cohomology of the dual Banach cochain complexes and vice versa. Therefore, certain results on bounded cohomology can be transferred to l^1-homology. For example, we obtain a new, simple proof of the fact that l^1-homology depends only on the fundamental group and that l^1-homology with twisted coefficients admits a description in terms of projective resolutions. In the second part, we study applications of l^1-homology concerning the simplicial volume of non-compact manifolds.

Stable URL: http://nbn-resolving.de/urn:nbn:de:hbz:6-37549578216

errata: A list of errata and comments to my doctoral thesis can be found here.

diploma thesis: The Proportionality Principle of Simplicial Volume

I wrote my diploma thesis under supervision of Prof. Dr. W. Lück.

Abstract. The aim of this diploma thesis is to give a full proof of the proportionality principle of simplicial volume, including a proof of the fact that (smooth) measure homology and singular homology (with real coefficients) are isometrically isomorphic.

The diploma thesis is also available at the arXiv: math.AT/0504106

errata:

• The manifold M in Corollary 5.10 has to be closed.
• Theorem 2.37 is only valid for connected locally finite CW-complexes.